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THE  AUTHOR 

elg  ©etu'cates  tfji's  Booh 

TO 

ALBERT    J.    SCIIERZER,    C.E., 

©ItJ  <£amratrc  anti  Scar  JFrtcntt, 

IX  TOKEX  OF   ESTEEM    FOJl   HIS   PIIOFESSIONAL 

ATTAINMENTS   AND    RESPECT   FOB  HIS 

-MAXL.Y   CHAKACTEIi. 


PEEFAOE. 


THE  author's  principal  aim  in  preparing  this  volume  has 
been,  as  its  title  indicates,  to  serve  that  large  class  of  young 
engineers  who,  like  himself,  have  not  had  the  advantage  of  a 
technical  education  before  going  out  for  their  livelihood. 

The  initial  chapters  are,  therefore,  given  to  a  compendious 
exposition  of  those  mathematical  truths  and  methods  which 
they  must  needs  become  familiar  with  from  the  beginning. 
Plane  Trigonometry,  Logarithms,  and  propositions  relating  to 
the  circle,  are  tools  of  the  craft  in  constant  use ;  ready  han- 
dling of  them  is  an  indispensable  condition  of  excellence.  Be 
not  discouraged  by  obscurities  and  difficulties  at  the  outset; 
light  will  gradually  break  on  the  scrutinizing  eye,  and  a  way 
always  open  to  manful  effort. 

These  chapters  are  followed  by  instructions  as  to  the  adjust- 
ment and  use  pf  instruments,  and  hints  concerning  field  rou- 
tine, which  it  is  thought  will  be  found  acceptable  to  the 
inexperienced  learner.  The  same  may  be  said  of  the  articles 
on  staking  out  work,  and  those  on  track  problems,  with  which 
the  text  of  the  book  closes.  They  have  been  written  with  the 
author's  own  early  ignorance  in  mind,  and  with  a  wish  to  set 
the  subjects  forth  as  plainly  as  possible,  disembarrassed  of 
hard  words  in  the  description,  and  of  unpractical  niceties  in 
the  operation. 

The  chapter  on  field  location  is  believed  to  include  all  the 
problems  likely  to  occur.  The  author,  in  compiling  it,  has 
taken  those  only  which  have  arisen  in  his  own  practice,  and 
which,  therefore,  may  arise  in  the  practice  of  others.  His 


vi  PREFA  CE. 

own  practice  having  been  unusually  large  and  diversified, 
probably  the  examples  given  will  prove  adequate,  directly  or 
indirectly,  to  all  contingencies. 

No  attempt  has  been  made  to  swell  the  bulk  of  the  volume 
with  imaginary  cases ;  the  object  being,  not  to  provide  barren 
mathematical  exercises,  but  to  teach  useful  knowledge. 

Problems,  also,  affecting  location  in  its  economical  aspects, 
—  the  balancing  of  physical  and  financial  conditions,  equating 
of  alternative  lines,  and  the  like,—  do  not  come  within  the 
scope  of  the  work,  and  are  therefore  not  treated. 

Considerable  pains  have  been  spent  on  the  tables.  However 
far  the  young  engineer  may  eventually  outgo  his  teacher  as  re- 
gards the  text  of  the  bx>k,  these  are  implements  of  his  art 
which  never  become  antiquated,  and  can  never  fall  into  dis- 
use. Those  herein  contained  which  are  original  will,  it  is 
hoped,  be  esteemed  worthy  of  place  with  their  well-approved 
associates. 

The  author  invites  friendly  criticism :  he  would  be  pleased 
to  receive  suggestions,  both  for  the  improvement  of  the  book, 
and  for  the  correction  of  possible  errors  in  it,  should  another 
edition  be  called  for. 

In  dismissing  the  work  from  his  hands,  the  precarious 
snatches  of  time  occupied  in  its  preparation,  by  day  and  by 
night,  during  the  past  two  years,  which  might  have  been  more 
agreeably  spent  in  reading,  talking,  or  musing,  recur  to  the 
writer's  mind;  and  the  thought  arises,  To  what  end  or  from 
what  motive  do  people  undertake  these  technical  labors?  Why 
should  Forney  and  Bourne  toil  to  simplify  steam  for  our  ap- 
prehension; Nystrom  to  compile  mechanical,  Molesworth  and 
Trautwine  to  epitomize  civil  engineering;  Henck  to  prepare 
his  elegant  manual  of  field  mathematics;  Box  to  illustrate 
hydraulics;  and  Shreve,  with  lucid  pen,  to  make  clear  for  us 
the  strains  in  truss  or  arch?  The  ordinary  motives  to  en- 
deavor here  have  no  place.  There  is  neither  fame  nor  profit 
in  these  drudging  enterprises.  At  best  the  author  gives  name 


PREFACE.  vii 

to  his  book;  he  remains  impersonal^ —  known  but  indirectly, 
and  but  to  a  class.  How,  then,  shall  we  account  for  his  labors? 
I  take  it,  the  Father  of  mankind  has  not  only  made  our  minds 
to  hunger  for  knowledge  as  our  bodies  for  food,  but  has  also 
imposed  upon  us  a  kindly  law  of  communion,  by  virtue  whereof 
we  cannot  do  otherwise,  without  violence  to  generous  nature, 
than  share  with  our  fellows  whatsoever  we  have  learned  that 
seems  new  and  useful.  Under  this  law  these  beneficial  works 
would  appear  to  have  had  their  being,  and  thus  pure  are  they 
from  the  stain  of  selfishness. 

Though  the  present  writer  would  not  arrogate  equal  fellow- 
ship in  the  eminent  brotherhood  named,  yet  he  may  justly 
claim  like  pureness  from  unworthy  motive,  and  certainly  feels 
like  comfort  at  heart  to  that  which  they  must  know,  for  having 
discharged,  in  what  measure  it  has  been  laid  upon  him,  the 

divine  obligation. 

WM.   F.  SIIUNK. 

,  N.J. 


ABBREVIATIONS. 


+    Increased  by. 

—    Diminished  by. 

X     Multiplied  by. 

-f-    Divided  by. 

=    Equal  to. 

*.  *    Since,  or  seeing  that. 

.*.     Hence,  or  therefore. 

I  Indicates  the  quotient  of  one  divided  by  the  other  of  the 
quantities  it  connects,  called  sometimes  the  ratio  of  the  quan- 
tities. 

; :  Indicates  an  equality  of  ratios,  and  connects  equal  ratios 
in  a  proportion.  Thus,  a  :  b  : :  c  :  d  indicates  that  a-S-b  =  c 
~  d;  or  it  may  be  read,  a  is  to  b  as  c  is  to  d. 

(  )  Brackets  indicate  that  the  operations  embraced  by  them 
shall  first  be  performed,  and  the  result  treated  as  a  single  factor 
in  the  remaining  processes  required  by  a  formula.  Thus, 
(a  X  6)  -r-  (a  -\-  b)  requires  that  the  product  of  a  and  b  shall  be 
divided  by  their  sum. 

A2.  A  small  secondary  figure  annexed  thus  to  an  expression 
is  called  its  exponent.  It  requires  the  principal  to  which  it  is 
attached  to  be  used  as  many  times  in  continued  multiplication 
as  there  are  units  in  the  exponent.  Thus,  A2  =  A  X  A ;  A3 
=  A  X  A  X  A,  which  is  called  the  cube,  or  third  power,  of  A. 

V  This  is  called  the  square  root  sign:  it  signifies  that  the 
square  root  of  the  quantity  covered  by  it  is  to  be  taken. 

V  If  preceded  by  a  small  secondary  figure,  called  the  index, 
as  in  the  marginal  figure,  it  indicates  that  the  cube  root  of  the 
quantity  covered  by  it  shall  be  taken ;  and  so  on. 

\j  If  the  index  be  fractional,  as  in  the  marginal  figure,  it 
requires  that  the  square  root  of  the  third  power  of  the  quantity 
covered  shall  be  taken. 

B.  M.  Bench-mark  :  any  fixed  reference  point  for  the  level, 

ix 


X  ABB11EYIA  TIONS. 

as  outcropping  ledge,  water-table  of  building,  or  other  perma- 
nent object.  Usually  a  blunt  conical  seat  for  the  rod,  hewn, 
on  a  buttressed  tree-base,  having  a  small  nail  sometimes  driven 
flush  in  the  top  of  it,  and  a  blaze  opposite,  oil  which  the  eleva- 
tion is  marked  with  kiel. 

T.  P.     Turning-point :  usually  marked  ©  in  the  field-book. 

P.  I.  Point  of  intersection:  as  of  tangents,  which  are  to  be 
connected  by  a  curve. 

A.  D.     Apex  distance:  i.e.,  the  distance  from  the  P.  I.  to 
the  point  where  a  curve  merges  in  the  tangent. 

P.  C.  Point  of  curve :  the  stake-mark  at  the  beginning  of 
a  curve. 

P.  T.  Point  of  tangent:  the  stake-mark  at  the  end  of  a 
curve. 

P.  C.  C.  Point  of  compound  curvature :  the  stake-mark 
where  a  curve  merges  in  another  of  different  curvature,  turn- 
ing in  the  same  direction. 

P.  R.  C.  Point  of  reverse  curvature :  the  stake-mark  where 
a  curve  merges  in  another  turning  in  the  opposite  direction. 

B.  S.     Backsight,  in  transit  work;  or  the  reading  of  the  rod 
to  ascertain  the  instrument  height  in  levelling. 

F.  S.  Foresight,  in  transit  work;  or  the  reading  of  the  rod 
to  ascertain  elevations  in  levelling. 

H.  I.  Height  of  instrument :  elevation  of  the  level  above 
the  datum  or  zero  plane. 

II.  W.     High  water. 

L.  "W.     Low  water. 


TABLE  OF  CONTENTS. 


PAGE 

LOGARITHMS. 

I.  Definitions  and  principles 3 

II.  Manner  of  using  the  tables 4 

To  find  the  logarithm  of  any  number          .        .  4 
To  find  the  number  corresponding  to  a  given 

logarithm 5 

Multiplication  by  means  of  logarithms       .        .  5 

Division  by  means  of  logarithms          ...  6 
To  raise  a  number  to  any  power  by  means  of 

logarithms 7 

To  extract  roots  by  means  of  logarithms    .        .  7 
PLANE  TRIGONOMETRY. 

III.  Definitions 11 

IV.  Natural  sines 12 

V.  Logarithmic  sines,  &c 13 

VI.  General  propositions      ......  15 

VII.  Solution  of  plane  triangles 16 

VIII.  Right-angled  plane  triangles         ....  18 
ADJUSTMENT  AND  USE  OF  INSTRUMENTS. 

IX.  General  remarks  on  adjustment  .        .        .        .23 

X.  The  level 24 

To  bring  the  intersection  of  the  cross-hairs  into 

the  optical  axis  of  the  telescope        ...  24 
To  bring  the  level  bubble  parallel  with  the  tele- 
scope axis 25 

To  adjust  the  wyes;  or,  in  other  words,  to  bring 
the  telescope  into  a  position  at  right  angles  to 

the  vertical  axis  of  the  instrument  ...  25 

XI.  Levelling 26 

Correction  for  the  earth's  curvature  and  refrac- 
tion          28 

To  find  differences  in  elevation  by  means  of  the 

barometer 29 

Heights  by  the  thermometer '.       ....  29 

xi 


Xii  TABLE    OF    CONTENTS. 

PAGE 

XII.  Setting  slope  stakes 30 

XIII.  Vertical  curves .        .  36 

XIV.  The  transit 40 

To  adjust  the  level  tubes 40 

To  adjust  the  vertical  hair  so  that  it  shall  re- 
volve in  a  plane,  and  mark  backsight  and  fore- 
sight points  in  the  same  straight  line       .        .  40 

To  adjust  the  needle 41 

XV.  Miscellaneous 42 

The  vernier 42 

To  read  an  angle 43 

To  re-magnetize  a  needle 44 

To  replace  cross-hairs 44 

To  fix  a  true  meridian 44 

PROPOSITIONS  AND  PROBLEMS  RELATING  TO  THE  CIRCLE. 

XVI.  Propositions  relating  to  the  circle        ...      49 
XVII.  Circular  curves  on  railroads         ....      50 
XVIII.  To   find  the    radius,   the    apex   distance,  the 

length,  the  degree,  &c.,  of  a  curve  ...      52 

Given  the  intersection  angle  I  and  radius  R,  to 
find  the  tangent  T 52 

Given  the  intersection  angle  I  and  tangent  T,  to 
find  the  radius  R 53 

Given  the  intersection  angle  I  and  chord  AB=  C, 
connecting  the  tangent  points,  to  find  the 
radius  R 54 

Given  the  intersection  angle  I  and  the  degree  of 
curvature  or  deflection  angle  D,  with  100-feet 
chords,  to  determine  the  length  of  the  long 
chord  C,  the  versed  sine  V,  the  external  secant 
S,  or  the  tangent  T 54 

Given  C,  V,  S,  or  T,  of  any  curve,  and  D,  the 
degree  of  curvature,  to  find  the  intersection 
angle  I 55 

Given  the  intersection  angle  I  and  deflection 
angle  D,  to  find  the  length  of  the  curve  .  .  55 

Given  any  radius  R  and  chord  C,  to  find  the  de- 
flection angle  D 57 

Given  any  radius  R  and  chord  C,  to  find  the  de- 
flection distance  d 57 

Given  any  radius  R  and  chord  C,  to  find  the 
tangential  angle  T 57 

Given  any  radius  R  and  chord  C,  to  find  the 
tangential  distance  t 58 


TABLE    OF    CONTENTS.  Xiii 

PAGE 

XIX.  Ordinates 58 

Given  any  radius  R  and  chord  C,  to  find  the 

middle  ordinate  M 58 

Given  the  radius  R,  chord  C,  and  middle  ordi- 
nate M,  to  find  any  other  ordinate    ...      59 
Ordinates  of  a  1°  curve,  chord  100-feet        .        .      60 
TRACING  CURVES  AND  TURNING  OBSTACLES  IN  THE  FIELD. 
XX.  To  trace  a  curve  on  the  ground  with  the  chain 

only .63 

XXI.  To  trace  a  curve  on  the  ground  with  transit  and 

100-feet  chain 66 

XXII.  Turning  obstacles  to  vision  in  tangent         .        .      71 

XXIII.  Turning  obstacles  to  measurement  in  tangent   .      73 
SUGGESTIONS  AS  TO  FIELD-WORK  AND  LOCATION-PROJECTS. 

XXIV.  Suggestions  concerning  field-work       ...      79 
XXV.  The  curve-protractor  and  the  projecting  of  loca- 
tions        84 

Table  showing  the  distance,  D,  in  feet,  at  which 
a  straight  line  must  pass  from  the  nearest 
point  of  any  curve  struck  with  radius  r,  in 
order  that  a  terminal  branch  having  a  radius 
R  =  2  r,  and  consuming  a  given  angle,  x,  may 
merge  in  said  straight  line  .  .  .  .  88 
Table  showing  the  distance,  d,  in  feet,  at  which 
curves  of  contrary  flexure  must  be  placed 
asunder,  in  order  that  the  connecting  tangent, 
T,  may  be  300  feet  long  .  .  .  .  .89 
PROBLEMS  IN  FIELD  LOCATION. 

XXVL  How  to  proceed  when  the  P.  C.  is  inaccessible  .      93 
XXVII.  How  to  proceed  when  the  P.  C.  C.  is  inaccessible,      95 
XXVIII.  To  shift  a  P.  C.  so  that  the  curve  shall  termi- 
nate in  a  given  tangent 96 

XXIX.  To  substitute  for  a  curve  already  located  one  of 
different  radius,  beginning  at  the  same  point, 
containing  the  same  angle,  and  ending  in  a 

fixed  terminal  tangent 97 

XXX.  Having  located  a  curve  A  B  C,  to  find  the  point 
B  at  which  to  compound  into  another  curve 
of  given  radius,  which  shall  end  in  tangent 
E  F,  parallel  to  the  terminal  tangent  of  the 
original  curve,  and  a  given  distance  from  it  .  98 
XXXI.  To  shift  a  P.  C.  C.  so  that  the  terminal  branch  of 

a  curve  shall  end  in  a  given  tangent         .        .      99 


[V  TABLE    OF    CONTENTS. 

PAGE 

XXXII.  Having  located  a  tangent,  A  B,  intersecting  a 
curve,  C  D,  from  the  concave  side,  to  find  the 
point  E  on  said  curve,  at  which  to  begin  a 
curve  of  given  radius  which  shall  merge  in 
the  located  tangent 102 

XXXIII.  Having  located  a  tangent,  A  B,  intersecting  a 

curve,  C  D,  from  the  convex  side,  to  find  the 
point  E  on  said  curve  at  which  to  begin  a  curve 
of  given  radius  which  shall  merge  in  the 
located  tangent .  103 

XXXIV.  To  locate  a  Y 103 

XXXV.  To  locate  a  tangent  to  a  curve  from  an  outside 

fixed  point 107 

XXXVI.  To  substitute  a  curve  of  given  radius  for  a  tan- 
gent connecting  two  curves        ....    108 
LXXVII.  To  run  a  tangent  to  two  curves  already  located  .    109 
RACK  PROBLEMS. 

XXVIII.  Reversed  curves 115 

XXXIX.  To  connect  two  parallel  tangents  by  a  reversed 

curve  having  equal  radii 115 

XL.  To  connect  two  parallel  tangents  by  a  reversed 

curve  having  unequal  radii        .        .        .        .117 
XLI.  A  reversed  curve  having  unequal  angles    .        .    119 
XLII.  A  reversed  curve  between  fixed  points        .        .    120 
XLIII.  To  connect  two  divergent  tangents  by  a  reversed 

curve 123 

XLIV.  To  shift  a  P.  R.  C.  so  that  the  terminal  tangent 

shall  merge  in  a  given  tangent    .        .        .        .125 
XLV.  To  pass  a  curve  through  a  fixed  point,  the  angle 

of  intersection  being  given         ....    127 

XL VI.  Frogs  and  switches 129 

To  find  the  radius  of  a  turnout  curve,  the  frog 
angles,  and  the  distances  from  the  toe  of  switch 

to  the  frog  points 129 

To  find  the  angle  of  switch-rail  with  main  track .    130 
To  find  the  distance  from  toe  of  switch  to  point 

of  main  frog 130 

To  find  the  radius  of  outer  rail  of  turnout  curve,    131 
To  find  the  main  frog  angle,  the  radius  of  outer 

rail  being  known 131 

To  find  the  angle  of  the  middle  frog  in  the  case 

of  a  double  turnout 131 

To  find  the  distance  from  toe  of  switch  to  point 
of  middle  frog       .......    131 

Turnout  tables         ...        .  135,  136 


TABLE    OF   CONTENTS.  X 

PAG 

XLVII.  To  locate  a  turnout lo 

XLVIII.  Crossings  on  straight  lines 13 

XLIX.  Crossings  on  curves 13 

L.  Elevation  of  the  outer  rail  on  curves  .        .        .  14 

Table  for  same 14 

LI.  Trackmen's  table  of  curves  and  spring  of  rails  .  14 

Explanation  of  same 14 

LIST  OF  TABLES. 

I.  Time  of  meridian  passage  of  North  Star  above  the 

pole  for  the  year  1870,  and  on 14 

II.  Time  of  extreme  elongation  of  North  Star  for  the 

year  1870,  lat.  40°,  and  on 1C 

III.  Azimuths  of  the  North  Star,  and  their  natural  tan- 

gents         1* 

IV.  Roods  and  perches  in  decimal  parts  of  an  acre  .        .  K 
V.  Decimals  of  an  acre  in  one  chain  length  of  100  feet, 

and  of  various  widths 1" 

VI.  Acres,  roods,  and  pert  lies  in  square  feet      .        .        .If 

VII.  Circular  arcs  to  radius  of  1 1' 

VIII.  Feet  in  decimals  of  a  mile II 

IX.  Inches  reduced  to  decimal  parts  of  a  foot   .        .        .  It 
X.  Radii  and  their  logarithms,  middle  ordinates,  and 

deflection  distances If 

XI.  Squares,  cubes,  &c.,  of  numbers  from  1  to  1042  .        .  1( 

XII.  Logarithms  of  numbers,  1  to  1000 11 

XIII.  Logarithmic   sines,    cosines,    tangents,  and   cotan- 

gents        1! 

XIV.  Natural  sines  and  tangents    .        .        .        .        .        .  & 

XVI.  Chords,  versed  sines,  external  secants,  and  tangents 

of  a  one-degree  curve 2( 

XV.  Slopes  for  topography 3] 

XVII.  Rise  per  mile  of  various  grades 3] 


LOGAEITHMS. 
I. -II. 


LOGAEITHMS. 


I. 

DEFINITIONS  AND  PRINCIPLES. 

1.  THE  logarithm  of  a  number  is  the  exponent  of  the  power 
to  which  it  is  necessary  to  raise  a  fixed  number  to  produce  the 
given  number ;  that  is  to  say,  it  represents  the  number  of  times 
a  fixed  number  must  be  multiplied  by  itself  in  order  to  produce 
any  given  number. 

The  fixed  number  is  called  the  base  of  the  system.  In  the 
common  system,  this  base  is  10. 

It  follows  from  the  above,  that  the  logarithm  of  any  power 
of  10  is  equal  to  the  exponent  of  that  power.  If,  therefore, 
a  number  is  an  exact  power  of  10,  its  logarithm  is  a  whole 
number. 

If  a  number  is  not  an  exact  power  of  10,  its  logarithm  will 
not  be  a  whole  number,  but  will  be  made  up  of  an  entire  part 
plus  a  fractional  part,  which  is  generally  expressed  decimally. 
The  entire  part  of  the  logarithm  is  called  the  characteristic ; 
the  decimal  part  is  called  the  mantissa. 

2.  The  characteristic  of  the  logarithm  of  a  whole  number 
is  positive,  and  numerically  1  less  than  the  number  of  places 
of  figures  in  the  given  number. 

Thus,  if  a  number  lies  between  1  and  10,  its  logarithm  lies 
between  0  and  1 ;  that  is,  it  is  equal  to  0  plus  a  decimal.  If  a 
number  lies  between  10  and  100,  its  logarithm  is  equal  to  1 
plus  a  decimal ;  and  so  on. 

3.  The  characteristic  of  the  logarithm  of  a  decimal  fraction 
is  negative,  and  numerically  1  greater  than  the  number  of  O's 
that  immediately  follow  the  decimal  point. 

The  characteristic  alone,  in  this  case,  is  negative,  the  man- 
3 


4  MANNER   OF  USING   THE   TABLES. 

tissa  being  always  positive.  This  is  indicated  by  writing  the 
negative  sign  over  the  characteristic :  thus,  2.380211  is  equiva- 
lent to  —  2  +  .380211. 

4.  The  characteristic  of  the  logarithm  of  a  mixed  number 
is  the  same  as  that  of  its  entire  part.     Thus  the  mixed  number 
74.103  lies  between  10  and  100;  hence  its  logarithm  lies  be- 
tween 1  and  2,  as  does  the  logarithm  of  74. 

5.  The  logarithm  of  the  product  of  two  numbers  is  equal  to 
the  sum  of  the  logarithms  of  the  numbers. 

The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  diminished  by  that  of  the  divisor. 

The  logarithm  of  any  power  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  multiplied  by  the  exponent  of  the  power. 

The  logarithm  of  any  root  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  divided  by  the  index  of  the  root. 

6.  The  preceding  principles  enable  us  to  abridge  labor  in 
arithmetical  calculations,  by  using  simple  addition  and  sub- 
traction instead  of  multiplication  and  division. 


II. 
MANNER  OF  USING  THE  TABLES. 

TO  FIND  THE  LOGARITHM  OF   ANY  NUMBER. 

1.  First  find  the  characteristic  by  rule  2,  3,  or  4,  given 
above. 

2.  Then,  if  the  number  be  less  than  100,  look  in  column  N 
of  the  table  for  10  times  or  100  times  the  amount  of  it ;  oppo- 
site this  multiple,  in  column  O,  will  be  found  the  mantissa. 

Thus  the  logarithm  of  6  is  0.778151 ;  that  of  84  is  1.924279. 

3.  If  the  number  lie  between  100  and  10000,  find  the  first 
three  figures  of  it  in  column  N;  then  pass  along  a  horizontal 
line  until  you  come  to  the  column  headed  with  the  fourth 
figure   of   the    number.     At    this   place   will    be   found    the 
mantissa. 

Thus  the  logarithm  of  7200  is  3.857332;   that  of  8536  is 
3.931254. 


MANNER   OF   USING    THE  TABLES.  5 

4.  If  the  number  be  greater  than  10000,  place  a  decimal  point 
after  the  fourth  figure,  thus  converting  the  number  into  a 
mixed  number.    Find  the  mantissa  of  the  entire  part  by  the 
method  last  given.     Then  take  from  column  D  the  correspond- 
ing tabular  difference,  multiply  this  by  the  decimal  part,  and 
add  the  product  to  the  mantissa  just  found.     The  principle 
employed  is  that  the  differences  of  numbers  are  proportional 
to  the  differences  of  their  logarithms,  when  these  differences 
are  small. 

Thus  the  logarithm  of  672887  is  5.827943;  that  of  43467  is 
4.638160. 

5.  If  the  number  be  a  decimal,  drop  the  decimal  point,  thus 
reducing  it  to  a  whole  number.     Find  the  mantissa  of  the  log- 
arithm of  this  number,  and  it  will  be  the  mantissa  required. 

Thus. the  logarithm  of  .0327  is  2.514548;  that  of  378.024  is 
2.577520. 

TO  FIND  THE   NUMBER  CORRESPONDING  TO   A  GIVEN 
LOGARITHM. 

6.  The  rule  is  the  reverse  of  those  just  given.     Look  in  the 
table  for  the  mantissa  of  the  given  logarithm.     If  it  cannot  be 
found,  take  out  the  next  less  mantissa,  and  also  the  corre- 
sponding number,  which  set  aside.     Find  the  difference  be- 
tween the  mantissa  taken  out  and  that  of  the  given  logarithm; 
annex  as  many  O's  as  may  be  necessary,  and  divide  this  result 
by  the  corresponding  number  in  column  D.     Annex  the  quo- 
tient to  the  number  set  aside,  and  then  point  off  from  the  left 
hand  a  number  of  places  of  figures  equal  to  the  characteristic 
plus  1 ;  the  result  will  be  the  number  required.     If  the  char- 
acteristic is  negative,  the  result  will  be  a  pure  decimal,  and 
the   number   of   O's    which  immediately  follow  the  decimal 
point  will  be  one  less  than  the  number  of  units  in  the  charac- 
teristic. 

Thus  the  number  corresponding  to  the  logarithm  5.233568 
is  171225.296;  that  corresponding  to  the  logarithm  2.233568  is 
.0171225. 

MULTIPLICATION  BY  MEANS  OF   LOGARITHMS. 

7.  Find  the  logarithms  of  the  factors,  and  take  their  sum; 
then  find  the  number  corresponding  to  the  resulting  logarithm, 
and  it  will  be  the  product  required. 


6  MANNER    OF   USING    THE   TABLES. 

Example. 
Find  the  continued  product  of  3.902,  597.16,  and  0.0314728. 

Operation. 

Log.  3.902.  .  .  0.591287 
Log.  597.16  .  .  .  2.776091 
Log.  0.0314728  .  2.497936 

1.865314  =  log.  73.3354,  the  product. 

Here  the  2  cancels  the  +  2,  and  the  1  carried  from  the  deci- 
mal part  is  set  down. 

DIVISION  BY  MEANS  OF  LOGARITHMS. 

8.  Find  the  logarithms  of  the  dividend  and  the  divisor,  and 
subtract  the  latter  from  the  former;   then  find  the  number 
corresponding  to  the  resulting  logarithm,  and  it  will  be  the 
quotient  required. 

Example  1. 
Divide  24163  by  4567. 

Operation. 

Log.  24163  .  .  .  4.383151 
Log.  4567  .  .  .  3.659631 

0.723520  =  log.  5.29078,  the  quotient. 

Example  2. 
Divide  0.7438  by  12.9476. 

Operation. 

Log.  0.7438.  .  .  1.871456 
Log.  12.9476  ...  1.112189 

2. 759267  =  log.  0.057447,  the  quotient. 

Here  1  taken  from  .  gives  2  for  a  result.     The  subtraction,  as 
in  this  case,  is  always  to  be  performed  in  the  algebraic  way. 

9.  The  operation  of  division,  particularly  when  combined 
with  that  of  multiplication,  can  often  be  simplified  by  using 
the  principle  of  the  arithmetical  complement. 

The  arithmetical  complement  of  a  logarithm  (written  a.  c.) 


MANNER   OF  USING   THE  TABLES.  7 

is  the  result  obtained  by  subtracting  it  from  10:  it  may  be 
written  out  by  commencing  at  the  left  hand,  and  subtracting 
each  figure  from  9  until  the  last  significant  figure  is  reached,^ 
which  must  be  taken  from  10.     Thus  8.130456  is  the  arithmet-* 
ical  complement  of  1.869544. 

To  divide  one  number  by  another  by  means  of  the  arith- 
metical complement,  find  the  logarithm  of  the  dividend  and 
the  arithmetical  complement  of  the  logarithm  of  the  divisor; 
add  them  together,  and  diminish  the  sum  by  10;  the  number 
corresponding  to  the  resulting  logarithm  will  be  the  quotient 

required. 

Example. 

Multiply  358884  by  5672,  and  divide  the  product  by  89721. 

Operation. 

Log.  358884    .     .     .     5.554954 

Log.      5672    .     .     .     3.753736 

(a.c.)Log.     89721   .     .     .     5.047106 

4.355796  =  log.  22688,  the  result. 

The  operation  of  subtracting  10  is  performed  mentally. 

TO  RAISE  A  NUMBER  TO  ANY  POWER    BY    MEANS    OF    LOGA- 
RITHMS. 

10.  Find  the  logarithm  of  the  number,  and  multiply  it  by 
the  exponent  of  the  power;  then  find  the  number  correspond- 
ing to  the  resulting  logarithm,  and  it  will    be    the  power 

required. 

Example. 

Find  the  5th  power  of  9. 

Operation. 

Log.  9 0.954243 

5 

4.771215  =  log.  59049,  the  power. 

TO  EXTRACT  ROOTS  BY  MEANS  OF    LOGARITHMS. 

11.  Find  the  logarithm  of  the  number,  and  divide  it  by  the 
index  of  the  root ;  then  find  the  number  corresponding  to  the 
resulting  logarithm,  aiid  it  will  be  the  root  required. 


8  MANNER  OF  USING   THE  TABLES. 

Example. 
Find  the  cube  root  of  4,096. 

Operation. 

Log.  4,096,  3.612360;  one-third  of  this  is  1.204120,  to  which 
the  corresponding  number  is  16,  which  is  the  root  sought. 

12.  When  the  characteristic  is  negative,  and  not  divisible  by 
the  index,  add  to  it  the  smallest  negative  number  that  will 
make  it  divisible,  and  then  prefix  the  same  number,  with  a 
plus  sign,  to  the  mantissa. 

Example. 

Find_the  4th  root  of  .00000081.  _The  logarithm  of  this  num- 
ber is  7.908485,  which  is  equal  to  8  +  1.908485,  and  one-fourth 
of  this  is  2.477121 ;  the  number  corresponding  to  this  logarithm 
is  .03:  hence  .03  is  the  root  required. 


PLANE   TEIGO1STOMETEY. 
III.-VIII. 


PLANE   TRIGONOMETRY. 


in. 

DEFINITIONS. 

1.  Plane  Trigonometry  treats  of  the  solution  of  plane  tri- 
angles. 

In  every  plane  triangle  there  are  six  parts,  —  three  sides  and 
three  angles.  When  three  of  these  parts  are  given,  one  being 
a  side,  the  remaining  parts  may  be  found  by  computation. 
The  operation  of  finding  the  unknown  parts  is  called  the  solu- 
tion of  the  triangle. 

2.  A  plane  angle  is  measured  by  the  arc  of  a  circle  included 
between  its  sides;  the  centre  of  the  circle  being  at  the  vertex, 
and  its  radius  being  1.     The  circle,  for  convenience,  is  divided 
into  360  equal  parts  called  degrees;    90  of  these  parts  are 
included  in  a  quadrant,  which  includes  one-quarter  of   the 
circle,  and  is  the  measure  of  a  right  angle.     Each  degree  is 
further  divided  into  60  equal  parts  called  minutes,  and  each 
minute  into  60  equal  parts  called  seconds.    Degrees,  minutes, 
and    seconds    are   de- 
noted by  the  symbols    T" ^— ?— ^ r' 

°,   ',  ":  thus  the  ex- 
pression 7°  22'  33"  is 

read,  7  degrees,  22 
minutes,  and  33  sec- 
onds. 

3.  The  complement 
of  an  angle  is  the  dif- 
ference between  that 
angle    and   a  right 
angle. 

4.  The  supplement  of  an  angle  is  the  difference  between 
that  angle  and  two  right  angles. 

11 


12  NAT  URAL    SINKS,   ETC. 

5.  Instead  of  employing  the  arcs  themselves,  certain  func- 
tions of  the  arcs  are  usually  employed,  as  explained  below.     A 
function  of  a  quantity  is  something  which  depends  upon  that 
quantity  for  its  value. 

The  .sine  of  an  angle  is  the  distance  from  one  extremity  of 
the  arc  enclosing  it,  to  the  diameter,  through  the  other  extrem- 
ity. Thus  P  M  is  the  sine  of  the  angle  M  O  A. 

The  cosine  of  an  angle  is  the  sine  of  the  complement  of  the 
angle.  Thus  NM  =  O  P  is  the  cosine  of  the  angle  M  O  A. 

The  tanyent  of  an  angle  is  a  right  line  which  touches  the 
enclosing  arc  at  one  extremity,  and  is  limited  by  a  right  line 
drawn  from  the  centre  of  the  circle  through  the  other  extrem- 
ity: the  sloping  line  which  thus  limits  the  tangent  is  called  the 
secant  of  the  angle.  A  T  is  the  tangent  and  O  T  the  secant  of 
the  angle  MO  A. 

The  versed  sine  of  an  angle  is  that  part  of  the  diameter  AP 
which  is  intercepted  between  the  foot  of  the  sine  and  the  ex- 
tremity of  the  enclosing  arc. 

The  cotangent  of  an  angle  is  the  tangent  of  the  complement 
of  that  angle:  the  co-versed  sine  and  cosecant  are  similarly 
defined.  Thus  BT',  BN,  and  OT'  are  respectively  the  co- 
tangent, co-versed  sine,  and  cosecant  of  the  angle  M  O  A. 

These  terms  are  in  practice  indicated  by  obvious  contractions; 
as,  sin.  A  for  the  sine  of  A,  cos.  A  for  the  cosine  of  A,  &c. 

6.  The  above  definitions  have  been  made  with  reference  to  a 
radius  of  1.     Any  function  of  an  arc  whose  radius   is  R   is 
equal  to  the  corresponding  function  of  an  arc  whose  radius  is 
1,  multiplied  by  the  radius  R.     So  also  any  function  of  an  arc 
whose  radius  is  1  is  equal  to  the  corresponding  function  of  ail 
arc  whose  radius  is  R,  divided  by  that  radius. 


IV. 

NATURAL  SINES,  ETC. 

1.  Natural  sines,  cosines,  tangents,  or  cotangents  are  those 
which  are  referred  to  a  radius  of  1.  They  may  be  used  for  all 
the  purposes  of  trigonometrical  computation;  but  it  is  found 
more  convenient,  in  many  cases,  to  employ  a  table  of  logarith- 
mic sines. 


LOGARITHMIC  SINES,  ETC.  13 


V. 

LOGARITHMIC   SINES,   ETC. 

1.  Logarithmic  sines,  cosines,  tangents,  or  cotangents  are  re- 
ferred to  a  radius  of  10,000,000,000,  of  which  the  logarithm  is 
10. 

TO  FIND  THE   LOGARITHMIC    FUNCTIONS    OF    AN    ARC    WHICH 
IS   EXPRESSED   IN   DEGREES   AND   MINUTES. 

2.  If  the  arc  is  less  than  45°,  look  for  the  degrees  at  the  top 
of  the  page,  and  for  the  minutes  in  the  left-hand  column; 
then  follow  the  corresponding  horizontal  line  till  you  come  to 
the  column  designated  at  the  top  by  sine,  cosine,  tang.,  or 
cotang.,  as  the  case  may  be;  the  number  there  found  is  the 
logarithm  sought. 

Thus,  log.  sin.    19°  55' ....     9.532812 
log.  tang.  19°  55' .     .     .     .     9.559097 

3.  If  the  angle  is  greater  than  45°,  look  for  the  degrees  at 
the  bottom  of  the  page,  and  for  the  minutes  in  the  right-hand 
column ;  then  follow  the  corresponding  line  towards  the  left, 
till  you  come  to  the  column  designated  at  the  bottom  by  sine, 
cosine,  tang,  or  cotang,  as  the  case  may  be ;  the  number  there 
found  is  the  logarithm  sought.     . 

Thus,  log.  cos.  52°  18'    .    .    .    .    9.786416  " 
log.  tan.  52°  18'    ....  10.111884 

4.  If  the  arc  is  expressed  in  degrees,  minutes,  and  seconds, 
proceed  as  before  with  the  degrees  and  minutes ;  then  multiply 
the  corresponding  number  taken  from  column  D  by  the  num- 
ber of  seconds,  and  add  the  product  to  the  preceding  result, 
for  the  sine  or  tangent,  and  subtract  it  therefrom  for  the  cosine 
or  cotangent. 

Example. 
Find  the  logarithmic  sine  of  40°  26'  28;/. 


12  NATURAL   SINE 8,  ETC. 

5.  Instead  of  employing  the  arcs  themselves,  certain  func- 
tions of  the  arcs  are  usually  employed,  as  explained  below.     A 
function  of  a  quantity  is  something  which  depends  upon  that 
quantity  for  its  value. 

The  sine  of  an  angle  is  the  distance  from  one  extremity  of 
the  arc  enclosing  it,  to  the  diameter,  through  the  other  extrem- 
ity. Thus  P  M  is  the  sine  of  the  angle  M  O  A. 

The  cosine  of  an  angle  is  the  sine  of  the  complement  of  the 
angle.  Thus  N  M  =  O  P  is  the  cosine  of  the  angle  M  O  A. 

The  tangent  of  an  angle  is  a  right  line  which  touches  the 
enclosing  arc  at  one  extremity,  and  is  limited  by  a  right  line 
drawn  from  the  centre  of  the  circle  through  the  other  extrem- 
ity: the  sloping  line  which  thus  limits  the  tangent  is  called  the 
secant  of  the  angle.  A  T  is  the  tangent  and  O  T  the  secant  of 
the  angle  MO  A. 

The  versed  sine  of  an  angle  is  that  part  of  the  diameter  AP 
which  is  intercepted  between  the  foot  of  the  sine  and  the  ex- 
tremity of  the  enclosing  arc. 

The  cotangent  of  an  angle  is  the  tangent  of  the  complement 
of  that  angle;  the  co-versed  sine  and  cosecant  are  similarly 
denned.  Thus  BT',  BN,  and  OT'  are  respectively  the  co- 
tangent, co-versed  sine,  and  cosecant  of  the  angle  M  O  A. 

These  terms  are  in  practice  indicated  by  obvious  contractions ; 
as,  sin.  A  for  the  sine  of  A,  cos.  A  for  the  cosine  of  A,  &c. 

6.  The  above  definitions  have  been  made  with  reference  to  a 
radius  of  1.     Any  function  of  an  arc  whose  radius  is  R  is 
equal  to  the  corresponding  function  of  an  arc  whose  radius  is 
1,  multiplied  by  the  radius  R.     So  also  any  function  of  an  arc 
whose  radius  is  1  is  equal  to  the  corresponding  fuiictioii  of  au 
arc  whose  radius  is  R,  divided  by  that  radius. 


IV. 

NATURAL  SINES,  ETC. 

1.  Natural  sines,  cosines,  tangents,  or  cotangents  are  those 
which  are  referred  to  a  radius  of  1.  They  may  be  used  for  all 
the  purposes  of  trigonometrical  computation;  but  it  is  found 
more  convenient,  in  many  cases,  to  employ  a  table  of  logarith- 
mic sines. 


LOGARITHMIC  SINES,   ETC.  13 


V. 

LOGARITHMIC   SINES,   ETC. 

1.  Logarithmic  sines,  cosines,  tangents,  or  cotangents  are  re- 
ferred to  a  radius  of  10,000,000,000,  of  which  the  logarithm  is 
10. 

TO  FIND  THE   LOGARITHMIC    FUNCTIONS    OF    AN    ARC    WHICH 
IS   EXPRESSED   IN   DEGREES   AND   MINUTES. 

2.  If  the  arc  is  less  than  45°,  look  for  the  degrees  at  the  top 
of  the  page,   and  for  the  minutes  in  the  left-hand  column; 
then  follow  the  corresponding  horizontal  line  till  you  come  to 
the  column  designated  at  the  top  by  sine,  cosine,  tang.,  or 
cotang.,  as  the  case  may  be;  the  number  there  found  is  the 
logarithm  sought. 

Thus,  log.  sin.     19°  55' ....     9.532312 
log.  tang.  19°  55' .     .     .     .     9.559097 

3.  If  the  angle  is  greater  than  45°,  look  for  the  degrees  at 
the  bottom  of  the  page,  and  for  the  minutes  in  the  right-hand 
column ;  then  follow  the  corresponding  line  towards  the  left, 
till  you  come  to  the  column  designated  at  the  bottom  by  sine, 
cosine,  tang,  or  cotang,  as  the  case  may  be ;  the  number  there 
found  is  the  logarithm  sought. 

Thus,  log.  cos.  52°  18'    .    .    .    .    9.786416  * 
log.  tan.  52°  IS'    .     .     .     .  10.111884 

4.  If  the  arc  is  expressed  in  degrees,  minutes,  and  seconds, 
proceed  as  before  with  the  degrees  and  minutes ;  then  multiply 
the  corresponding  number  taken  from  column  D  by  the  num- 
ber of  seconds,  and  add  the  product  to  the  preceding  result, 
for  the  sine  or  tangent,  and  subtract  it  therefrom  for  the  cosine 
or  cotangent. 

Example. 
Find  the  logarithmic  sine  of  40°  26'  28". 


16 


SOLUTION  OF  PLANE  TRIANGLES. 


VII. 
SOLUTION  OF  PLANE  TRIANGLES. 

1.  It  is  usually,  though  not  always,  best  to  work  the  propor- 
tions  in  trigonometry   by  means  of   logarithms,   taking  the 
logarithm  of  the  first  term  from  the  sum  of  the  logarithms  of 
the  second  and  third  terms,  to  obtain  the  logarithm  of  the 
fourth  term;  or  adding  the  arithmetical  complement  of  the 
logarithm  of  the  first  term  to  the  logarithms  of  the  other  two, 
to  obtain  that  of  the  fourth. 

2.  There  are  three  distinct  cases  in  which  separate  rules  are 
required. 

CASE  I. 

3.  When  a  side  and  an  angle  are  two  of  the  given  parts,  the 
solution  may  be  effected  by  proposition  2  of  the  preceding 
section. 

If  a  side  be  required,  say,  — 

As  the  sine  of  the  given  angle  is  to  its  opposite  side, 

So  is  the  sine  of  either  of  the  other  angles  to  its  opposite  side. 

4.  If  an  angle  be  required,  say,  — 

As  one  of  the  given  sides  is  to  the  sine  of  its  opposite  angle, 
So  is  the  other  given  side  to  the  sine  of  its  opposite  angle. 
The  third  angle  becomes  known  by  taking  the  sum  of  the 
two  former  from  180°. 

Example  1. 

Given  angle  A  =  24°  26';  angle 
=  36°  43';  side  6  =  137.6:  to  find 
side  a. 

_»u 

As  sin.  B  .  .  log.  9.776598 
Is  to  sin.  A  .  log.  9.616616 
So  is  b  .  .  .  log.  2.138618 

11.755234,  sum  of  2d  and  3d  terms, 
To  a,  95.2  .     .    log.    1.978636  less  1st  term. 


SOLUTION  OF  PLANE  TRIANGLES.  •         17, 

Example  2. 
Given,  sides  a  and  &,  as  above,  and  angle  A;  to  find  angle  B. 

As  side  a (a.  c.)  log.    8.021364 

Is  to  sin.  A     .......      log.    9.616616 

So  is  side  b log.    2.138618 

To  sin.  B  =  36°  43' log.    9.776598  sum. 

CASE  II. 

5.  When  two  sides  and  the  included  angle  are  given,  the 
solution  may  be  effected  by  means  of  propositions  3  and  4. 
Thus,  take  the  given  angle  from  180°;  the  remainder  will  be 
the  sum  of  the  other  two  angles. 

Then,  by  proposition  3,  — 

As  the  sum  of  the  given  sides  is  to  their  difference, 

So  is  the  tangent  of  half  the  sum  of  the  remaining  angles 
to  the  tangent  of  half  their  difference. 

Half  the  sum  of  the  remaining  angles  added  to  half  their 
difference  will  give  the  larger  of  them,  and  half  their  sum 
diminished  by  half  their  difference  will  give  the  lesser  of  them. 

The  solution  may  be  completed  either  by  proposition  4,  or 
by  proposition  2,  as  in  Case  I. 

Example. 

Given  side  a  =  95.2,  side  6  =  137.6,  and  the  included  angle 
c  =  H8°  51';  to  find  the  remaining  angles.  Here  180.00  — 
118°  51' =  61°  09',  the  sum  of  the  remaining  angles. 

As  sum  of  given  sides,  232.8 log.  2.366983 

Is  to  their  difference,      42.4 log.  1.627366 

So  is  tang.  •}  sum  of  rem.  angles,  30°  34J;    .     log.  9.771447 

To  tang.  I  their  difference  =  6°  08$'    .    .    .    log.  9.031830 

Adding  half  the  difference  to  half  the  sum,  30°  34^  +  6° 
OS}' =36°  43',  =  the  larger  angle,  B.  Deducting  half  the 
difference  from  half  the  sum  =24°  26'  =  the  smaller  angle,  A. 

This  case  is  susceptible  of  solution  also  by  means  of  propo- 
sition 6. 


18  RIGHT-ANGLED   PLANE   TRIANGLES. 


CASE  TIT. 

6.  When  the  three  sides  of  a  plane  triangle  are  given,  to 
find  the  angles. 

First  Method. 

Assume  the  longest  of  the  three  sides  as  base ;  then  say, 
conformably  with  proposition  5,  — 

As  the  base  is  to  the  sum  of  the  two  other  sides, 

So  is  the  difference  of  those  sides  to  the  difference  of  the 
segments  of  the  base. 

Half  the  base  added  to  half  the  said  difference  gives  the 
greater  segment,  and  diminished  by  it  gives  the  less;  thus,  by 
means  of  the  perpendicular  from  the  vertical  angle,  the  original 
triangle  is  divided  into  two,  each  of  which  falls  under  the  first 
case.  Or  they  may  be  solved  by  the  simpler  methods  applica- 
ble to  right-angled  triangles. 

Second  Method. 

7.  Find  any  one  of  the  angles  by  means  of  proposition  6,  and 
the  remaining  angles  either  by  a  repetition  of  the  same  rule, 
or  by  the  relation  of  sides  to  the  sines  of  their  opposite  angles. 


VIII. 
RIGHT-ANGLED  PLANE  TRIANGLES. 

1.  Right  angles  may  be  solved  by  the  rules  applicable  to  all 
plane  triangles;  and  it  will  be  found,  since  a  right  angle  is 
always  one  of  the  data,  that  the  rule  usually  becomes  simplified 
in  its  application. 

2.  When  two  of  the  sides  are  given,  the  third  may  be  found 
by  means  of  the  rule  that  the  square  of  the  hypothenuse  is 
equal  to  the  sum  of  the  squares  of  the  remaining  sides. 

3.  Another  method  for  solving  right-angled  triangles   is  as 
follows:  — 

To  find  a  side.     Call  any  one  of  the  sides  radius,  and  write 
upon  it  the  word  "  radius."     Observe  whether  the  other  sides 


RIGHT-ANGLED  PLANE  TRIANGLES.  19 

become  sines,  tangents,  cosines,  or  the  like,  and  write  upon 
them  the  proper  designations  accordingly.     Then  say, 
As  the  name  of  the  given  side  is  to  the  given  side, 
So  is  the  name  of  the  required  side  to  the  required  side. 

4.  To  find  an  angle.     Assume  one  side  to  be  radius,  and 
mark  the  remaining  sides  as  before.     Then  say, 

As  the  side  made  radius  is  to  radius, 
So  is  the  other  given  side  to  the  name 
of  that  side; 
Which  determines  the  opposite  angle. 

5.  Applying  this  method  to  the  right- 
angled   triangle  ABC,   and   calling  the 
hypothenuse  a  radius,  we  shall  have, 


c  =  a  sin.  C  -f-  R ;  hence  sin.  C  =  Re  -f-  a. 
b  =  a  cos.  C  -r-  R;  hence  cos.  C  =  Rb  -f-  a. 

Then,  assuming  the  side  b  to  be  radius,  we  shall  have, 
c  =  b  tang.  C  -f-  R;  hence  tang.  C  =  Re  -j-  b. 

If  radius  be  called  1,  the  natural  sines  and  cosines  will  be 
used  in  the  application  of  these  formulas;  they  are  often  more 
convenient  than  logarithms  in  railroad  practice,  especially 
when  the  numbers  which  measure  the  sides  of  the  triangle  are 
either  less  than  12,  or  are  resolvable  into  factors  less  than  12. 


ADJUSTMENT  AND    USE 

OF 

INSTRUMENTS. 
IX.-XV. 


ADJUSTMENT  AND    USE. 

OF 

INSTRUMENTS. 


IX. 

GENERAL  REMARKS  ON  ADJUSTMENTS. 

1.  Care  should  be  taken  in  all  instrumental  adjustments, 
where  screws  work  in  pairs,  to  loosen  one  before  tightening  its 
opposite. 

2.  Remember  that  the  eye-piece  inverts  the  image  of  the 
cross-hairs,  and  that  consequently  any  movement  of  it,  by 
means  of  the  small  capstan  head  screws  on  the  outside  of  the 
telescope-barrel,  should  be  in  the  direction  which  would  seem 
to  increase  the  error  requiring  correction. 

3.  Before  beginning  the  adjustments,  screw  the  object-glass 
close  home,  and  make  a  pin-scratch  across  its  rini  and  the  end 
of  the  tube,  by  which  to  mark  its  proper  place;  draw  out  the 
eye-piece  until  the  cross-hairs  are  exactly  in  focus;  that  is  to 
say,  until  no  movement  of  the  eye  shall   appear  to  displace 
them,  and  bring  the  object  to  be  observed  clearly  into  view. 

4.  Never  permit  the  glasses  to  be  rubbed  with  a  gritty  fabric. 
To  remove  the  dust  from  them,  use  a  soft,  clean  handkerchief, 
and  change  often  the  part  applied. 

23 


24  THE  LEVEL. 


X. 

THE  LEVEL. 

TO    BRING    THE     INTERSECTION     OF     THE     CROSS-HAIRS     INTO 
THE   OPTICAL   AXIS    OF   THE    TELESCOPE. 

1.  Set  the  instrument  firmly,  cast  loose  the  wyes,  and,  by 
levelling  and  tangent  screws,  bring  either  of  the/cross-hairs  to 
coincide  with  a  well-defined  object,  distant  from  400  to   000 
feet,  or  as  much  farther  as  distinct  vision  can  be  had  free  from 
heat  ripple.     Gently  rotate  the  telescope  half-way  around  in 
the  wyes.    If  the  cross-hair  selected  for  treatment  then  fails 
to  coincide  with  the  object,  reduce  the  error  one-half  by  moans 
of  the  small  capstan  head  screws  at  right  angles  to  it  on  the 
telescope-barrel.     Bring  the  spider-line  again  to  coincide  with 
the  object  by  means  of  the  levelling  and  tangent  screws,  and, 
if  necessary,  repeat  the  operation.     Proceed  in  the  same  man- 
ner with  the  other  cross-hair.     If  the  error  is  large,  bring  both 
nearly  right  before  undertaking  their  final  adjustment. 

2.  Having  thus  adjusted  the  line  of  collimation  upon  a  dis- 
tant point,  requiring  the  object-tube  to  be  drawn  well  in,  select 
a  point  close  by,  which  shall  require  it  to  be  thrust  out  almost 
to  its  limit.     If  any  error  appears,  correct  half  of  it  with  the 
small  screws  provided  for  the  purpose,  a  little  forward  of  the 
diaphragm,   and  usually  protected   by  a  movable  sleeve    on 
the  outside;  correct  the  other  half  with  the  levelling-screws. 
After  completing  this   adjustment,  test  the  former  one  on  a 
distant  object,  and,  if  necessary,  repeat  the  operations. 

3.  In  the  transit,  the  small  guide-ring  screws  used  for  this 
adjustment  are  covered  by  the  bulb  of  the  cross-bar  in  which 
the   telescope  is  fixed,  and   are   therefore   inaccessible.     The 
adjustment,  however,  is  one  not  liable  to  become  deranged  in 
either  instrument,   and,   in  the  transit,  is  of  comparatively 
small  importance. 

4.  The  young  practitioner  should  bear  in  mind   that  the 
intersection  of  the  cross-hairs  may  coincide  with  the   optical 
axis  of  the  telescope,  and  yet  be  out  of  centre  as  regards  the 
field  of  view.     Such  eccentricity  does  not  affect  the  working 
accuracy  of  the  instrument,  which  depends  upon  the  position 


THE  LEVEL.  25 

of  the  object-piece  solely.     It  may  be  removed  by  manipulation 
of  the  small  screws  securing  the  inner  end  of  the  eye-piece. 

TO  BRING    THE   LEVEL   BUBBLE    PARALLEL   WITH   THE    TELE- 
SCOPE  AXIS. 

5.  Clamp  the  instrument  over  either  pair  of  levelling  screws, 
and  bring  the  bubble  to  the  middle  of  its  tube.  Turn  the  tele- 
scope slightly  on  its  bearings,  so  that  the  bubble-case  shall 
project  a  little  on  one  side  or  the  other.  If  the  bubble  slips, 
correct  half  its  movement  by  means  of  the  small  lateral  capstan 
head  screws  at  one  .end  of  the  case.  Return  the  telescope  to 
its  first  position,  level  up  again,  and  repeat  the  operation 'until 
the  erroneous  movement  ceases.  This  adjustment  brings  the 
telescope  and  level  into  the  same  vertical  plane. 

C.  Next,  the  bubble  being  at  the  middle  of  its  tube,  carefully 
lift  the  telescope  out  of  the  wyes,  turn  it  end  for  end,  and 
replace  it.  If  the  bubble  settles  away  from  the  middle,  bring 
it  half-way  back  by  means  of  the  capstan-heads,  working  up 
and  down  at  one  end  of  the  case.  Again  middle  it  with  the 
levelling  screws,  and  repeat  the  operation  until  the  error  is 
corrected. 

TO  ADJUST  THE  WYES  ;  OR,  IN  OTHER  WORDS,  TO  BRING  THE 
TELESCOPE  INTO  A  POSITION  AT  RIGHT  ANGLES  TO  THE 
VERTICAL  AXIS  OF  THE  INSTRUMENT. 

7.  Close   the  wyes.     Unclamp.     Set  the  telescope  directly 
over  two  of  the  levelling  screws,  and  with  them  bring  the 
bubble  to  the  middle  of  the  tube.     Then  rotate  the  telescope 
horizontally,    until   it  stands  over  the  same  pair  of  screws, 
changed  end  for  end.     If  the  bubble  errs,  correct  one-half  of 
the  deviation  with  the  capstan  head  nuts  at  the  end  of  the 
bar,  and  one-half  with  the   levelling  screws.    Place  the  tele- 
scope over  the  other  pair  of  levelling  screws.     Repeat  the 
operation  there;  and  continue  the  corrections,  over  one  and 
the  other  pair  of  levelling  screws  alternately,  until  the  bubble 
stands  without  varying  during  an   entire  revolution  of  the 
instrument  upon  its  vertical  axis. 

8.  The  capstan  head  nuts  on  the  cross-bar  should  be  moved 
by  gradual  stress,  not  by  pounding.     They  are  a  rude  device. 
With  so  short  a  leverage  as  the  length  of  the  common  adjust- 
ing-pin supplies,  it  is  almost  impossible  to  give  them  a  smooth, 


26 


LEVELLING. 


manageable  motion.  They  reproach  the  instrument-maker's 
art  as  unchecked  hydrophobia  and  cancer  do  that  of  medicine, 
or  mercenary  villany  that  of  law,  and  should  be  supplanted  by 
better  practice. 

9.  Having  thus  completed  the  principal  adjustments  in  their 
proper  order,  bring  the  telescope  and  its  bubble-case  as  nearly 
vertical  in  the  wye  bearings  as  hand  and  eye  can  make  them, 
and  by  reference  to  a  plumb-line,  or  the  corner  of  a  well-built 
house,  see  if  the  vertical  hair  is  out  of  true.  If  so,  slightly 
loosen  two  opposite  screws  of  the  diaphragm,  and  correct  the 
error  by  turning  it.  Try  again  the  adjustment  of  the  line  of 
colhmation  before  pinning  up  the  wyes. 


XI. 

LEVELLING. 

1.  Suppose  O  the  starting-point;  1,  2,  3,  &c.,  the  stakes  of 
survey;  and  A  the  initial  bench-mark.  Wherever  convenient 
the  elevation  of  A  above  mean  tide  should  be  ascertained. 
It  is  to  be  regretted  that  this  was  not  done  from  the  outset, 


under  statute  provisions  requiring  maps  and  profiles  also  to 
be  filed  at  the  several  State  capitals.  In  that  case,  not  only 
would  much  after  labor  and  expense  by  way  of  duplicate  sur- 
veys have  been  spared,  but  the  older  Commonwealths  would 
now  have  in  hand  materials  for  the  preparation  of  physio- 
graphical  maps,  the  value  of  which  to  science,  to  the  engineer, 
and  to  the  economical  geologist,  it  were  hard  to  measure. 


LEVELLING. 


27 


2.  For  the  purposes  of    a  railroad-survey,  however,  such 
determination  is  not  needful.     Any  elevation  may  be  assumed 
for  A,  taking  care  only  that  it  be  large  enough  to  avoid  the 
possibility  of  having  minus  levels,  which  would  be  inconven- 
ient.   Zero  of  the  datum  should  be  below  the  lowest  probable 
ground  on  the  contemplated  line. 

3.  Let  the  elevation  of  the  initial  bench-mark,  A,  in  the 
figure,  be  taken  at  -j-200.    Set  the  level  at  13,  and  suppose  the 
rod  on  the  B  M  to  read  2.22.     The  "  instrument  height "  then 
is  202.22.     If  the  rod  at  sta.  O  reads  8.4,  the  elevation  at  that 
point  is  202.22  —  8.4  ==  193.8.     The  rod  being  1.9  at  sta.  1, 
the  elevation  there  is  202.2  —  1.9  =  200.3.    If  desirable  to  turn 
at  sta.  2,  drive  a  pin  nearly  to  the  ground  at  that  stake;  sup- 
pose the  rod  on  it  to  read  0.81.    The  elevation  then  is  202.22  — 
0.81  =  201.41.    Now  move  the  instrument  to  0,  and,  sighting 
back  to  sta.  2,  let  the  rod  standing  on  the  pin  read  2.64.     This 
makes  the  new  "  instrument  height"  at  C  =  201.41,  the  height 
of    sta.  2,  -f  2.64  =  204.05,  and  the  elevations  at  3,  4,  5,  or 
other  points  observed  from  C   are  found  by  deducting  the 
"  rods"  at  those  points  from  the  ascertained  instrument  height 
at  the  new  point  of  observation. 

4.  It  thus    appears    how  simple  is  the  rule  of   levelling, 
namely:  Find  the  "instrument  height"  by  adding  the  "back- 
sight" to  the  elevation  of  the  point  upon  which  the  rod  stands 
for  that  purpose:  from  the  "instrument  height"  thus  found 
deduct  the  "foresights,"  severally,  in  order  to  find  the  eleva- 
tions of  the  points  at  which  such  foresights  are  taken. 

5.  The  foregoing  example  would  appear  in  the  field-book  as 
follows :  — 


STA. 

B.  S. 

INST. 

F.  S. 

ELEVA. 

REMARKS. 

BM 

200.00 

B  M  on  W.  Oak. 

2  22 

202!  22 

40  ft.  N.  of  Sta.  O. 

0 

. 

sM 

193.8 

1 

m 

1.9 

200.3 

2 

, 

0.81 

201.41 

.  . 

2  64 

204.05 

3 

g 

Si  7 

206!  3 

4 

, 

3.2 

200.8 

5 

• 

10.36 

193.69 

0.  Iii  levelling  where  great  exactness  is  necessary,  the  rod  at 
turning-points  should  be  read  to  thousandths,  and  the  reading 
checked  by  J,he  leveller.  Before  taking  it  down,  after  clamp- 


28  LEVELLING. 

ing  the  target  fast,  it  should  be  swayed  slowly  to  and  fro  in  the 
direction  of  the  instrument  to  make  sure  of  getting  the  full 
height.  In  foul  weather  the  rodnian  should  take  care  that 
the  foot  of  the  rod  does  not  ball  up  with  mud  or  snow.  The 
leveller  should  have  his  cross-hairs  free  from  parallax,  the  tar- 
get in  focus,  and  see  his  bubble  true  at  the  moment  of  obser- 
vation. He  should  also  set  the  instrument  about  half-way 
between  turning-points  when  practicable,  balancing  largely 
unequal  sights  by  subsequent  ones  similarly  unequal  in  the 
opposite  direction;  and  his  turning-points,  even  on  favorable 
ground,  ought  not  to  be  more  than  600  or  800  feet  asunder. 

7.  On  ordinary  railroad  field  work  such  nicety  as  is  implied 
in  most  of  these  rules  is  not  required.     To  read  to  the  nearest 
tenth  is  sufficient,  especially  where  the  progress  of  the  party 
depends  in  a  good  degree  on  the  level;  as,  for  example,  in  run- 
ning grade  lines  on  preliminary  survey.     The  location  levels 
are  usually  carried  along  more  carefully;  but  even  then  the 
writers  practice  has  been  to  turn  to  hundredths  only. 

8.  The  Philadelphia  Rod  is  the  best  for  our  service.    The 
sliding  halves  are  unconnected  except  by  brass  sleeves    or 
clips,  which  guide  them,  and  are  therefore  not  liable  to  bind  in 
wet  weather.     They  are  made  by  William  J.  Young's  Sons, 
who  some  years  ago,  at  the  writer's  suggestion,  supplied  what 
seemed  to  be  their  only  defect  by  adopting  rivets  for  fastening 
the  clips  instead  of  wood  screws:  the  screws  had  a  tendency  to 
work  loose  in  the  field,  and  cause  the  parts  to  chafe  or  jam. 
These  rods  are  clearly  figured,  so  as  to  be  legible  at  a  distance 
of  several  hundred  feet ;  the  leveller  is  thus  enabled  to  take 
intermediate  elevations  rapidly,  and,  when  necessary,  to  do 
his  work  with  the  aid  of  an  unlettered  rodnian. 


9.    CORRECTION   FOR  THE  EARTH'S   CURVATURE  AND   REFRAC- 
TION. 

The  correction  for  a  100-feet  "station"  is  .000215;  for  one 
mile,  0.6.  It  is  to  be  added  to  the  calculated  elevation  of  the 
point  observed,  or  to  be  deducted  from  the  "rod"  before 
calculating  the  elevation,  in  the  case  of  a  long  unbalanced 
sight.  It  varies  as  the  square  of  the  distance.  Calling  the 
required  correction  A,  for  any  given  distance  D,  then  A  = 
.000215  X  D-  if  D  is  in  "stations,"  and  A  =  0.6  X  D  -  if  D 
is  in  miles.  Thus  the  correction  for  10  stations  would  be 


LEVELLING.  29 

.0215;  for  50  stations,  0.5375;  for  10  miles,  60  feet,  and  a  spire 
or  tree  top  apparently  level  with  the  instrument  at  that  dis- 
tance would  really  be  GO  feet  above  it.  Transposing  the  equa- 
tion we  have  D  =  Y/  A-f-0.6.  In  this  form  it  is  applicable  to 
the  determination  of  distances  at  sea.  The  Peak  of  Teneriffe, 
for  example,  16,QOO  feet  high,  should  be  just  visible  from  the 
sea-level  at  a  distance  =  ^10000-4-0.0  =  say  163  miles. 

10.   TO  FIND  DIFFERENCES   IN   ELEVATION   BY  MEANS  OF  THE 
BAROMETER, 

Call   the  required   difference  D;   the   barometrical  reading 
at  the  lower  stand,  L;  that  at  the  upper  stand,  U. 

Then,  D  =  [  (L  —  U)  -i-  (L  +  U)  I  X  55000. 

Example. 
L  =  26.64;  U  =  20.82. 

Then,  L  — U=  5.82        ....     log.  0.764923 
=  47.46        ....     lo.  1.676328 


0.1226 Diff.  —1.088595 

And  0.1226  X  55000  =  6743,  the  required  difference  of  elevation 
in  feet. 

11.  A  closer  approximation  is  thought  to  be   attainable  by 
using  a  thermometer  in  connection  with  the  mercurial  barome- 
ter.    In  thai  case,  having  found  the  difference  as  above,  add 
•^1^  of  the  result  for  each  degree  by  which  the  mean  tempera- 
ture of  the  air  at  the  two  stands  exceeds  55°;  subtract  the  like 
proportion  if  the  mean  temperature  be  below  55°.     When  the 
upper  thermometer  reads  highest,  for  "subtract"  say  "add," 
and  vice  versa  in  the  foregoing  rule. 

12.  The  naked  formula,  however,  will  usually  be  sufficient 
for  the  engineer.     He  can  prescribe  gradients  by  it  for  surveys, 
which  shall  develop  the  ground  to  be  occupied,  and  can  decide 
between  summits  well  differenced  in  height.     If  not  so  differ- 
enced, questions  of  detour,  of  approaches,  and  the  like,  will 
contribute  to  determine  the  expediency  of  making  an  instru- 
mental examination. 

13.    HEIGHTS  BY  THE  THERMOMETER. 

T  =  the  difference,  in  degrees  Fahrenheit,  between  212°,  the 
temperature  of  boiling  water  at  the  sea  level,  and  that  at  the 
place  of  observation. 


30  SETTING  SLOPE  STAKES. 

H  =  the  height  of  place  of  observation  above  or  below  the 
sea  in  feet. 

H  =  513  T  -f  T2. 

Example. 

T  =  212°  —  208°  =  4°. 
H  =  (513  X  4)  -f  4-  =  2068  feet. 


XII. 
SETTING  SLOPE  STAKES. 

1.  Like  swallowing,  this  is  more  easily  done  than  described. 
To  no  detail  of  field  service  does  the  proverb  more  fitly  apply, 
that  "work  makes  the  workman." 

2.  The  problem  is,  to  find  where  a  formation  slope  of  given 
inclination,  beginning  at  the  side  of  the  road-bed,  must  needs 
intersect  the  ground  surface.     Formation  slopes  are  usually 
stated  in  parts  horizontal  to  one  part  vertical.     Thus  a  slope 
of  45°  is  "  1  to  1."     A  slope  of  "  2  to  1 "  has  a  horizontal  reach 
of  two  feet  to  each  foot  vertical.     The  carriages  of  a  stairway 
with  twelve-inch  treads  and  eight-inch  risers  would  have  a 
slope  of  "H  to  1." 

3.  To  fix  the  point  where  any  proposed  formation  slope  must 
meet  the  surface  on  level  ground,  is  quite  simple;  the  distance 
from  the  centre  line  being  obviously  made  up  of  half  the  width 
of  road-bed  added  to  the  horizontal   distance  due  from  the 
slope,  to  the  depth  of  cut  or  height  of  fill.     Thus,  with  20  f^et 
road-bed,  9  feet  cut,  and  slope  of  1£  to  1,  the  distance  out 
would  be  10  -f-  9  -f-  4£  =  23£  feet,  as  shown  in  the  annexed 
diagram. 


4.  On  slant  or  broken  ground,  the  solution  is  more  difficult: 
recourse  must  then  be  had  to  the  level,  with  a  rodman,  a  tape- 
man,  and,  for  good  speed,  an  axeman  to  assist. 


SETTING   SLOPE  STAKES. 


31 


Example  No.  1. 

5.  Let  the  accompanying  figure  represent  the  cross-section  at 
any  given  point  of  a  proposed  excavation  ;  road-bed  20  feet  wide, 
cutting  at  centre  stake  12  feet,  and  formation  slopes  1  to  1. 


--------  26.0  ------------  *| 


6.  The  first  step  is  to  set  the  level,  as  at  A,  commanding,  let 
us  suppose,  the  lower  slope,  and  to  ascertain  its  height  above 
grade  at  the  proposed  section.     This  is  usually  done  by  refer- 
ence to  the  nearest  bench,  and  pegging  from  stake  to  stake  as 
the  work  progresses.     Unless  the  ground  is  very  steop,  and  the 
slope-stakes  largely  different  in  elevation,  labor  will  be  saved 
and  likelihood  of  error  reduced  by  levelling  over  the  centre 
line  beforehand,  as  a  separate  job,  and  marking  on  centre 
stakes  the  cuts,  fills,  and  grade  points,  that  is  to  say,  the  points 
where  excavation  passes  into  embankment.     The  rods  should 
be  taken  carefully  at  the  stakes,  and  the  latter  marked  on 
their  backs  to  the  nearest  tenth,  as  "  grade,"  "  C  12,"  signify- 
ing cut  12  feet,  or  "F  6.2,"  signifying  Jill  6.2  feet,  for  ex- 
ample.    This  being  done,  each  centre  stake  serves  as  a  bench- 
mark for  slope  staking  at  that  section,  and  each  cross  section 
can  be  staked  out  independently. 

7.  Instrument  height,  in  the  example  treated,  being  by  either 
method  fixed  at  15.5  above  grade,  the  next  step  is  a  guess  how 
far  out  from  the  centre  stake  the  formation  slope  would  proba- 
bly meet  the  ground  surface.     The  closeness  of  the  guess  will 
correspond  to  the  experience  and  natural  skill  of  the  leveller: 
the  young  engineer  should  not  be  discouraged  if  he  misses  the 
mark  rather  widely  in  his  early  trials. 


32  SETTING   SLOPE 

8.  It  is  true,  that,  on  a  uniform  declivity,  he  might  aid  con- 
jecture by  taking  a  rod  distant  half  the  width  of  road-bed,  or 
10  feet,  from  the  centre  stake,  ascertain  thus  the  slope  of  the 
ground  surface  as  well  as  the  cutting  at  that  point;  and  with 
these  data,  knowing  also  the  formation  slope,  approximate 
the  point  sought  by  solving  the  terminal  triangle  of  the  pro- 
posed section,  indicated  by  dotted  lines  in  the  figure.  But,  in 
practice,  he  will  find  it  tlie  quicker  and  better  way  to  approxi- 
mate the  point  by  vividly  imagining  the  underground  forma- 
tion lines;  or  by  vividly  imagining  a  level  section,  the  upper 
surface  of  which  shall  coincide  with  his  instrument  height, 
15.5  feet  above  grade  This  gives  him  a  point  in  the  air, 
10  -f-  15.5  =  25.5  feet  out  from  the  centre  stake,  level  with  the 
instrument,  as  the  limit  of  the  imaginary  section;  and  from 
that  point  he  can  pretty  well  judge  where  a  line  corresponding 
to  the  formation  slope  must  meet  the  ground. 

i).  Suppose  him,  by  either  method,  or  even  by  random  guess, 
to  think  that  10  feet  for  half  the  road-bed,  and  10  more  for 
the  slope,  looks  about  right.  The  formation  slope  being  1  to 
1,  this  implies  a  cutting  of  10  feet  at  the  side  stake,  and  a  rod, 
therefore,  of  15.5  —  10.0  =  5.5  feet.  Taking  a  rod  accordingly, 
20  feet  out,  measured  horizontally  from  the  centre  stake,  he 
finds  it  to  be  11.0  instead  of  5.5,  indicating  that  he  has  gone 
too  far  down  hill.  Let  him  now  reason  that  the  rod  of  11.0 
corresponds  to  a  cutting  of  15.5  —  11.0  =  4.5  feet,  and  that  a 
cutting  of  4.5  feet  corresponds  to  a  distance  out  of  10  -f-  4.5 
=  14.5  feet.  Try,  then,  a  rod  14.5  feet  out.  It  proves  to  be 
9.0,  corresponding  to  a  cutting  of  15.5  —  9.0  =  6.5,  instead  of 
4.5  feet,  and  a  distance  out  of  16.5  instead  of  14.5  feet.  Try, 
next,  16.5  feet  out;  the  rod  there,  of  10.0  instead  of  9.0,  shows 
him  again  to  be  in  error  on  the  down-hill  side  of  his  object; 
but  the  lessening  error  shows  also  that  he  is  approaching  it, 
and  that  a  few  more  like  trials  will  reach  it. 

10.  Recurring  to  his  first  error  with  the  11.0  feet  rod,  he 
cannot  fail  to  observe  after  a  little  practice,  since  the  ground 
ascends  thence  toward  the,  centre  line,  that  the  side  stake 
must  fall  farther  out  than  the  point  where  his  secolid  trial  was 
made;  that  its  true  position,  in  fact,  divides  the  distance  be- 
tween those  points  of  observation  into  two  parts  which  are  to 
one  another  directly  as  the  inclinations  of  the  formation  slope 
and  the  ground  surface.  By  degrees  he  will  grow  skilful  in 
divining  this  true  position,  and,  becoming  meanwhile  quick  in 


SETTING   SLOPE  STAKES. 


33 


observation,  will  place  a  slope  stake  on  the  second  or  third 
trial,  without  conscious  effort  of  mind. 

11.  Next,  suppose  the  level  at  13,  25.5  feet  ahove  grade,  com- 
manding the  upper  slope. 

Note  the  change  of  ground  11  feet  out,  and  take  a  rod  there, 
recording  the  observation.  The  cutting  at  that  point  is 
25.5  —  U.5  =  16  feet,  corresponding  to  a  distance  out  for  the 
side  stake  of  10  -f-  16  =  26  feet,  if  the  ground  were  level.  A 
trial  rod  26  feet  out  reads  7.8,  corresponding  to  a  cutting  of 
25.5  —  7.8  =  17.7  feet,  and  a  distance  out  for  the  side  stake 
of  10  -f-  17.7  =  27.7  feet,  showing  that  the  point  sought  is  still 
beyond.  A  repetition  of  such  trials  will  finally  fix  it;  but,  as 
in  the  case  of  the  lower  slope,  practice  will  speedily  lessen  the 
'number  and  abridge  the  labor  of  them. 

12.  The  foregoing  section  would  be  noted  in  the  field  book 
as  follows:  — 


STA. 

Dis. 

LEFT. 

CENTRE 

RIGHT. 

AREA. 

C.YDS 

258 

50 

+  5.8 

iO 

+  12.0 

+  16.0 
11.0 

+  18.0 
28.0 

Example  No.  2. 

13.  In  the  annexed  figure,  representing  an  embankment  14 
feet  wide  on  top,  with  side  slopes  of  H  to  1,  the  first  thing  to 
attract  attention  is  that  the  instrument  is  1  foot  below  grade, 


and  that,  therefore,  1.0  is  to  be  added  to  all  rods,  in  order  to 
Imd  the  height  of  embankment  above  the  points  at  which  rods 
are  takon. 

14.  Consider  the  down-hill   side.     The  engineer,  with  the 
ground  ill  view,  and  with  the  height  of  embankment  at  the 


34 


SETTING  SLOPE  STAKES. 


centre  stake  to  aid  him  in  forming  an  airy  image  of  the  pro- 
posed section,  judges  that  the  natural  surface  and  the  forma- 
tion slopes  will  meet  30  feet  out.  Of  this  distance,  7  feet  are 
due  to  half  the  road-bed,  and  23  feet  to  horizontal  reach  of  the 
embankment  slope.  The  slope  being  1£  to  1,  or  f,  the  hori- 
zontal reach  of  23  feet  corresponds  to  a  vertical  height  of 
f  of  23  =  15.3  feet;-  and,  since  the  instrument  is  1  foot  he- 
low  grade,  to  a  rod  at  the  supposed  embankment  base  of 
153  —  1.0  =  14.3  feet.  But  the  rod  at  that  point  is  only  11 
feet,  to  which,  if  1  foot,  the  distance  of  instrument  below 
grade,  be  added,  the  height  of  embankment  would  be  12  feet. 
He  may  then,  as  in  the  case  of  the  upper  slope  in  Example  No, 
1,  try  a  rod  at  the  distance  out  corresponding  to  the  11  feet- 
rod,  or  12  feet  embankment.  This  distance  would  be  7  +  12 
-J-  6  =  25  feet,  where,  on  trial,  the  rod  proves  to  be  10  feet, 
instead  of  11  feet,  corresponding  to  an  embankment  height  of 
10  -f-  1  =  11  feet,  and  to  a  distance  out  of  7  -f  11  +  5.5  =  23.5 
feet.  Approximating  thus,  by  shorter  and  shorter  steps,  he 
jfinally  reaches  the  point  sought. 

I  15.  The  process  in  fixing  the  upper  slope  stake  is  similar  to 
that  used  in  fixing  the  lower  one  in  Example  No.  1.  The 
iseveral  steps  are  designated  by  small  letters  in  the  figure,  and 
a  detail  of  them  is  not  thought  necessary. 

16.  This  section  would  be  noted  in  the  field  hook  as  fol- 
lows:— 


STA. 

Dis. 

LEFT. 

CENTRE 

RIGHT. 

AREA. 

C.Ycs. 

140 

62 

—  9.4 

22.  G 

—  6.3 

—  3.2 

12.7 

Example  No.  3. 

17.  Here  is  a  case,  partly  in  rock  excavation,  slope  \  to  1; 
partly  in  embankment,  slope  1^  to  1;  road-bed  17  feet  wide,  9 
feet  of  which  are  on  the  right  of  the  centre  line  and  8  feet  on 
the  left. 

18.  For  the  lower  slope  suppose  the  instrument  height  at  A 
to  be  G.5  feet  above  grade ;  centre  cutting  2.5  feet.     Find  first, 
with  a  G.5  feet  rod,  the   grade  point,  to  left  of  centre   line, 
which    prmcs   to   be  2.5   foot   out.     Note  it,  and  set  a  stake 
there  marked  "grade."     Note  also  the  change  of  ground  5.5 


SETTING  SLOPE  STAKES.  85 

feet  out  and  10.0  —  0.5  ==  3.5  feet  below  grade.     Then  set  the 
lower  slope  stake  as  in  Example  No.  2,  observing  that  in  this 


case  the  instrument  is  above  grade,  and  that  its  height  above 
grade  is  to  be  deducted  from  the  rod  at  any  point  in  order  to 
obtain  the  height  of  grade  above  snch  point. 

19.  Move  the  instrument  to  B,  say  22.5  feet  above  grade*. 
This  elevation,  if  the  cutting  on  that  side  be  deemed  to  equal 
it,  corresponds  to  a  distance  out  of  9  feet  for  road-bed  added  to 
(22.5  -7-4)  for  slope;  total,  14.0  feet.    The  trial  rod,  however, 
at  that  distance,  instead  of  reading  0,  reads  6  feet,  indicating  a 
cut  22.5  —  0.0  =  10.5  feet  deep,  and  a  distance  out  correspond- 
ing thereto  of  9.0  +  (16.5  -f-  4)  =  13.1  feet.     Trying  again  at 
this  distance  out,  the  rod  reads  7.6  instead  of  0  feet,  requiring 
a  further  movement  towards  the  centre   line  of  (7.0  —  6)  -4-  4 
=  0.4  feet.     Thus  by  approximations  much  more  rapid  than  in 
the  case  of  a  flatter  formation  slope,  the  point  is  soon  fixed. 

20.  The  field  record  of  the  above  is  as  follows:  — 


STA. 

Dis. 

LEFT. 

(,'ENTKE 

RIGHT. 

AKEA. 

C.Yus. 

0.0  ^ 

328 

40 

—  6.9- 

2.5  i 
-3.5  f 

+  2.5 

4-15.0 

18.3 

12.8 

5.5  J 

VERTICAL   CURVES. 


XIII. 
VERTICAL  CURVES. 

DIAGRAM  GIVING  THE  ORDINATES  OF  A  PARABOLA  AT  IN- 
TERVALS OF  ^V  TO  THE  SPAN»  THE  MIDDLE  OKDINATE 
BEING  UNITY. 

F 


^x 

^ 

^^" 

X^ 

r-f 

^~~ 

1  



> 

"S 

r-^ 

/ 

+          .t 

7         «e 

9 

Xie       *a 

o      .* 

4          .C 

5          .« 

Q         '7 

5          '8 

3          '-£ 

'12           II               1 

o 

o 

1.  Suppose  gradients  descending  right  and  left  at  an  equal 
rate  from  the  summit  B,  and  that  it  is  required  to  truncate  the 
summit  with  a  vertical  curve  extending  150  feet  each  way. 

A  circular  arc  consuming  so  small  an  angle  may  be  treated 
as  a  parabola,  in  which  the  external  secant  BF  is  equal  to. the 
versed  sine  FD.  Referring  to  the  above  diagram,  ordinates  4 
and  8  will  be  seen  to  correspond  to  the  ordinates  between 


chord  AC  and  the  curve  in  this  instance,  which  ordinates 
therefore  will  be  equal  to  the  middle  ordinatc  FD  multiplied 
by  0.89  and  0.55  respectively.  Adding  these  multiples  to  the 
grade  elevation  at  A,  the  elevations  of  the  intermediate  points 
selected  will  be  ascertained. 


VERTICAL    CURVES, 


87 


Example  1. 

Elevation  at  A  =  -f  94.0;  A  B  =  +  1  in  100;  B  C  =  —  1  in 
100;  AD,  D  C,  each  =  150  feet  or  1.5  stations  of  100  feet  each. 

Hence  BD  ==  1.5;  and  FD  =  0.75  feet. 

Ordinate  8  =  0.75  X  0.55  =  0.41. 

Ordinate  4  =  0.75  X  0.89  =  0.67. 
Elevation  of  grade  at  8  —  8  =  94.0  -f-  0.41  =  94.41. 
Elevation  of  grade  at  4  —  4  =  94.0  +  0.67  =  94.67. 
Elevation  of  grade  at  D  =  94.0  +  0.75  =  94.75. 

Example  2. 


Elevation  at  A  =  +  94.0.  A  B  =  -f  1  in  100;  B  C  =  —  0.4 
in  100 ;  A  H,  level ;  A  D,  D  H,  each  =  200  feet,  or  2  stations, 
divided  into  50  feet  spaces,  the  points  of  division  correspond- 
ing therefore  to  ordinates  3,  6,  and  9  of  the  preceding  diagram. 

C II  =  1  X  2  —  0.4  X  2  =  2.0  —  0.8  =  1.2  feet. 


Ascent  from  A  to 
8  =  0.15  per  50  feet. 


C  along  chord  AC  =  CH  -=-  8  =  1.2  -f- 


BE  =  BD  — I  CH  =  2  —  0.6  =  1.4. 
.'.  FE  =  1.4-f-2  =  0.7. 

Ordinates  at  9  —  9  =  0.7  X  0.44  =  0.31. 
Ordinates  at  6  —  6  =  0.7  X  0.75  =  0.52. 
Ordinates  at  3  —  3  =  0.7  X  0.94  =  0.66. 
Mid-ordinate  =  0.70. 

The  elevations  then  along  the  chord  AC,  ascending  at  the 
rate  of  0.15  per  50  feet,  will  be :  — 

A963          0369          C 
94.0  94.15  94.30  94.45  94.60  94.75  94.90  95.05  95.20 


38 


VEK  TIC  A  L    CUR  YES. 


to  which  add  the  ordinates  just  found:  — 

0.0     0.31     0.52     O.C6     0.70     0.66     0.52     0.31     0.0 
and  the  grade  elevations  on  the  curve  will  be:  — 

94.0  94.46  94.82   95.11  95.30  95.41  95.42  95.36  95.2 

Example  3. 

Elevation  at  A  =  +  94.0 ;  A  B  =  -f  1  in  100 ;  B  C,  A II,  level. 
AD,  BC,  each  200  feet  divided  into  50-feet  spaces,  the  points 


of  division  corresponding  therefore  to  ordinates  3,  6,  and  9  of 
the  ordinate  diagram  C  H  =  1  X  2  =  2  feet. 

Ascent  from  A  to  C  along  chord  A  C  =  C  H  -f-  8  =  0.25  per 
50  feet. 

BE  =  B  I)  —  &  C  II  =  1  foot. 

.VFE  =  1  -f-2  =  0.5. 
Ordinates  9  —  9  =  0.5  X  0.44  =  0.22. 
Ordinates  6  —  6  =  0.5  X  0.75  =  0.37. 
Ordinates  3  —  3  =  0.5  X  0.94  =  0.47. 
Mid.ordinale  =  =  0.50. 

The  elevations  then  along  the  chord  A  C,  ascending  at  the 
rate  of  0.25  per  50  feet,  will  be:  — 

A9         6         3          0         3         6         9         C 
94.0  94.25  94.5     94.75  95.0     95.25  95.5     95.75  96.0 

to  which  add  the  ordinates  just  found:  — 

0.0     0.22     0.37     0.47     0.5       0.47     0.37     0.22     0.0 
And  the  grade  elevations  on  the  curve  will  be:  — 

94.C  94.47  94.87  95.22  95.5     95.72  95.87  95.97  96.0 


VER  TJCAL   CUR  VES. 


39. 


Example  4. 

Elevation  at  A  =  +  04.0;  AB  =  —  1  in  100;  BC,  AH, 
level;  AD,  BC,  each  150  feet,  divided  into  50-feet  spaces,  the 
points  of  division  corresponding  therefore  to  ordinates  8  and  4 
of  the  initial  diagram  C  II  =  1  X  1.5  =  1.5. 


Descent  from  A  to  C  along  chord 


CH-f-6  =  0.25. 


E  B  =  D  B  —  D  E  =  1.5  —  0.75  =  0.75 

.-.  FE  =  0.75-1-2  =  0.375 
Ordinates  8  —  8  =  0.375  X  55  =  0.21 
Ordinates  4  —  4  =  0.375  X  89  =  0.33 
Mid  ordiuate  =  0.37 

The  elevations  then  along  the  chord  A  C,  descending  at  the 
rate  of  0.25  per  50  feet,  will  be:  — 

A       8          4          0          4          8          C 
94.0  93.75  93.5     93.25  93.0     92.75  92.5 

From  which  deduct  the  ordinates  just  found, 

0.0     0.21     0.33     0.37     0.33     0.21     0.0 
And  the  grade  elevations  on  the  curve  will  be:  — 

94.0  93.54  93.17  92.88   92.67  92.54  92.5 

The  figures  are  drawn  much  distorted,  in  order  to  make  the 
illustration  clear. 

2.  With  profile  paper  at  hand,  a  convenient  and  quite  suf- 
ficient determination  of  the  grade  elevations  on  a  vertical 
curve  may  be  made  by  drawing  the  gradients  to  a  scale  of  2 
feet  to  an  inch  vertical,  and  50  feet  to  an  inch  horizontal.  By 
means  of  the  curve  protractor  (Art.  XXV.  1  )  a  suitable  arc  may 
then  be  lilted  and  struck  in,  and  the  elevations  read  off. 


40  THE  TRANSIT. 


XIV. 
THE  TRANSIT. 

1.  Should  the  vernier  and  circle  plates  be  out  of  parallel, — 
should  one  or  the  other  be  sprung,  a  defect  shown  by  a  slight 
rocking  motion  when  the  rims  are  pinched  alternately  on  op- 
posite sides,  —  the  instrument  must  be  sent  to  the  shop  for 
repair.  This  is  a  common  disease  of  transits  in  their  old  age: 
instrument-makers  need  to  study  its  cause  and  cure. 


2.  TO  ADJUST  THE  LEVEL  TUBES. 

Bring  the  bubbles  to  the  middle  of  them  by  means  of  the 
levelling  screws.  Turn  the  top  of  the  instrument  horizontally 
half  way  round.  If  the  bubbles  then  err,  reduce  the  error  one- 
half  with  the  small  adjusting  screws  attached  to  the  tubes, 
and  one-half  with  the  levelling  screws.  Repeat  until  the  ad- 
justment is  perfect. 


3.  TO  ADJUST  THE  VERTICAL  HAIR  SO  THAT  IT  SHALL  RE- 
VOLVE IN  A  PLANE,  AND  MARK  BACKSIGHT  AND  FORE- 
SIGHT POINTS  IN  THE  SAME  STRAIGHT  LINE. 

Try,  first,  by  reference  to  the  corner  of  a  well-built  house, 
or  to  a  plumb-line,  whether  the  hair  be  truly  vertical.  If  it  is 
not,  loosen  the  four  small  capstan  head  screws  on  the  outside 
of  the  barrel  slightly,  and  gently  tap  the  topmost  one  right  or 
left,  until  the  adjustment  is  effected. 

4.  Then,  after  bringing  the  four  screws  to  a  snug  bearing 
again,  direct  the  cross-hair  to  the  edge  of  some  well-defined 
object,  as  a  chain  pin,  or  stake,  placed  400  or  600  feet  distant. 
Upset  the  telescope,  and  place  a  like  mark  at  about  the  same 
distance,  and  level  in  the  opposite  direction.  Unclamp.  Re- 
volve the  instrument  horizontally  on  its  spindle  half  way 
round,  and  direct  the  cross-hair  to  the  point  first  observed. 
Again,  upset  the  telescope.  If  the  cross-hair  now  strikes  aside 
from  the  second  mark,  correct  one-quarter  of  the  error  by 
means  of  the  lateral  capstan  head  screws  on  the  barrel,  and 


THE   TRANSIT.  41 

one-quarter  with  the  tangent  screw.  Repeat  until  the  adjust- 
ment is  effected.  An  experienced  transitman  will  generally 
prefer  to  make  this  adjustment  without  aid,  points  in  range 
being  readily  found. 

5.  Having  thus  brought  the  cross-hair  to  revolve  in  a  plane, 
it  is  next  to  be  seen  whether  the  plane  in  which  it  revolves  is 
truly  vertical.  To  do  so,  set  the  instrument  near  the  base  of 
some  lofty  point,  as  a  church  spire  or  chimney,  on  which  point 
direct  the  cross-hair,  and  then,  tilting  the  object  end  of  the 
telescope  downwards,  set  a  pin,  or  make  a  pencil  dot  in  line. 
Unclamp  the  spindle;  turn  the  instrument  horizontally  half- 
way round;  clamp  fast;  fix  the  cross-hair  again  on  the  lower 
point,  and  try  it  on  the  upper  one.  If  it  misses,  correct  half 
the  error  by  means  of  the  adjusting  screws  now  usually  pro- 
vided, atone  of  the  bearings  of  the  cross-bar;  or,  if  these  be 
lacking,  by  filing  off  the  feet  of  the  standard  which  supports 
the  higher  end  of  the  cross-bar. 

6.    TO   ADJUST  THE   NEEDLE. 

Having  removed  the  cap,  and  placed  the  instrument  con- 
veniently in  a  still  room,  push  one  end  of  the  needle  a  little 
aside  from  the  point  where  it  tends  to  settle,  and  exactly  to 
some  figured  division  line  on  the  graduated  circle.  There 
gently  slay  it  in  position  by  means  of  a  small  woodcif  block, 
an  ivory  die,  or  the  like.  Observe  where  the  opposite  end 
strikes.  If  between  graduation  lines,  mark  the  precise  spot 
with  a  sharp  pencil.  Turn  the  needle  end  for  end,  and  stay 
the  reverse  point  at  the  division  line  first  observed.  Again 
spot  with  the  pencil  where  the  opposite  end  stakes.  Midway 
of  these  two  pencil  spots  make  another.  Take  the  needle  off 
the  pivot,  and  bend  it  this  way  or  that,  until,  by  repeated 
trials,  when  replaced  with  one  end  stayed  at  the  division  line 
first  observed,  the  other  shall  cut  the  midway  pencil  spot. 

7.  The  needle  being  thus  straightened,  proceed  to  rectify  the 
position  of  the  centre  pin,  if  necessary,  by  bending  it  with  nip- 
pers so  that  the  needle  shall  cut  opposite  degrees. at  the  quarter 
points  of  the  circle. 


42 


MIS  CEL  LA  NEO  US. 


XV. 

MISCELLANEOUS. 

THE  VEHXIER. 

1.  The  vernier  in   the   transit  is  a  short 
graduated  arc,  movable  around  the  graduated 
circle  of  the  instrument,  by  means  of  which 
subdivisions  of  the  circle  graduation  can  be 
read.     There  are  many  varieties  of  the  ver- 
nier; but  a  knowledge  of  the  principle  upon 
which  one  is  made  introduces  the  student  to 
an  easy  acquaintance  with  all. 

2.  Suppose  the  tenth  part  of  a  foot  to  be 
marked  off  on  a  straight  edge  into  ten  equal 
parts,  and  that  on  another  straight  edge  a 
space  equal  in  length  to  nine  of  these  parts  is 
divided  also  into  ten  equal  parts.     The  sub- 
divisions of  the  latter  scale  will  then  each  be 
nine-tenths  as  large  as  the  subdivisions  of  the 
former;  and  if  the  graduated  edges  are  placed 
together,  with  the  zero  marks  in  both  exact- 
ly lined,  thl  first  mark  of  the  latter,  or  ver- 
nier, scale  will  fall  short  of  the  first  mark  of 
the  former,  or  limb,  so  to  speak,  by  one-tenth 
part  of  the  first  space  on  the  limb;  that  is  to 
say,  by  one-tenth  part  of  one-hundredth  of 
a  foot,  or  one-thousandth  of  a  foot.     The  sec- 
ond mark  of  the  vernier  will  fall  short  of  the 
second  mark  of  the  limb  by  two-thousandths 
of  a  foot,  and  so  on.     If,  therefore,  the  ver- 
nier scale  be  moved  slowly  forward,  the  suc- 
cessive oppositions  of  the  scale  marks  will 
indicate  successive  advances  of  the  vernier, 
each  equal  to  the  one-thousandth  part  of  a 
foot.    The  marginal  example  reads  0.217  =  six 
feet,  two-tenths^  one-hundredth,  and  seven- 
thousandths. 

3.  The  annexed  figure  represents  the  transit 


MIS  CEL  LANEO  US. 


43 


vernier,  together  with  a  part  of  the  graduated  circle.  The 
vernier  is  a  double  one,  for  con- 
venience in  reading  angles  right 
or  Jeft.  It  will  be  observed  that 
a  space,  equal  to  twenty-nine  half 
degrees  on  the  limb,  is  laid  off 
from  zero  each  way  on  the  ver- 
nier, and  there  subdivided,  on 
both  sides  of  zero,  into  thirty 
equal  parts.  If  now  the  zeros 
are  brought  into  line,  the  first 
marks  of  the  vernier  right  and 
left  will  fall  one-thirtieth  part  of 
a  half  degree  short  of  the  first,  or 
half-degree  marks  on  the  limb; 
that  is  to  say,  one  minute  short. 
The  vernier,  therefore,  is  scaled 
to  read  minutes;  and,  if  its  zero 
mark  be  moved  slowly  half  a 
degree  on  the  limb,  its  several 
subdivision  marks,  one  after  an- 
other in  arithmetical  succession, 
will  be  seen  to  line  with  marks 
of  the  limb  until  the  thirtieth  is 
reached,  when  zero  will  be  found 
to  have  traversed  the  half  degree 
space. 

4.   TO  READ   AN  ANGLE. 

First  note  whether  the  vernier 
has  been  moved  right  or  left; 
then  observe  on  the  limb  the 
number  of  full  degrees,  and  the 
half-degree,  if  any,  which  zero  of 
the  vernier  has  passed;  next, 
look  along  the  vernier  from  its 
zero  towards  the  right,  if  the 
movement  has  been  towards  the 
right,  and  from  zero  towards  the 
left,  if  the  movement  has  been 

towards  the  left,  until  a  "minute"  mark  is  found  exactly  in 
line  with  some  mark  on  the  limb.     Add  the  number  of  that 


44  MISCELLANEOUS. 

minute  mark  on  the  vernier  to  the  angle  already  ascertained 
within  half  a  degree  from  the  limb:  the  sum  will  be  the  angle 
sought.  The  vernier  in  the  figure  reads  1°  20'  L. 

5.  In  some  respects  a  vernier  graduated  decimally  would  be 
more  convenient  on  railroad  locations,  where  the  100-feet  chain 
is  used;  the  calculation  of  engineers'  tables  to  sixtieths  of  a 
degree  has  prevented  its  adoption. 

6.    TO   RE-MAGNETIZE   A   NEEDLE. 

Lay  the  north  half  flat  on  a  smooth,  hard  surface,  and 
with  gentle  pressure  draw  tlto  south  pole  of  a  common  magnet 
over  it,  from  the  centre  outwards,  withdrawing  the  magnet 
from  it  six  or  eight  inches  after  each  pass.  Repeat  ten  or  a 
dozen  times.  Treat  the  south  half  of  the  needle  in  the  same 
manner  with  the  north  pole  of  the  magnet.  Replace  the  bal- 
ancing wire.  If  the  needle  yet  proves  to  be  sluggish,  take  out 
the  centre  pin,  and  newly  point  and  polish  it. 

7.  If  the  needle,  by  reason  of  electricity,  clings  to  the  cover- 
ing glass  in  the  field,  a  touch  of  the  moist  finger  to  the  top  of 
the  cover  will  release  it. 

8.  Do  not  suffer  idlers  to  play  it  about  with  knives,  keys, 
and  the  like. 

9.  When  the  instrument  is  out  of  use,  leave  the  needle  free. 


10.  TO   REPLACE   CROSS-HAIRS. 

Take  out  the  eye-glass  tube.  Remove  the  small  lateral 
capstan  head  screws  which  hold  the  cross-hair  ring  athwart 
the  barrel.  Loosen  the  vertical  screws,  and,  taking  care 
throughout  to  observe  the  position  of  the  ring,  in  order  that  it 
may  be  got  back  again  right  side  up  and  right  face  forward, 
turn  it  lengthwise  of  the  barrel.  Insert  the  end  of  a  pine 
sliver  into  one  of  the  side  holes,  take  out  the  vertical  screws, 
and  withdraw  the  ring.  Stretch  across  new  hairs,  in  the  scores 
traced  for  them,  of  the  finest  clean  spider-line;'  secure  them 
with  a  touch  of  gum  or  wax,  and  put  the  ring  in  by  a  reverse 
process. 

11.  TO   P'lX   A   TRUE   MERIDIAN. 

By  equal  shadows  of  the  sun. 

On  level  ground  or  ice,  set  up  a  pole.     Two  or  three  hours 


MISCELLANEOUS.  45 

• 

before  noon,  mark  the  extremity  of  its  shadow.  With  radius 
reaching  to  that  mark,  from  a  centre  on  the  surface  vertically 
below  the  top  of  pole,  strike  an  arc  eastward.  Two  or  three 
hours  after  noon,  watch  for  the  moment  when  the  extremity 
of  the  shadow  touches  the  arc.  There  make  another  mark. 
The  true  meridian  will  pass  from  the  centre  midway  between 
the  two  marks,  if  the  observations  be  made  about  the  period  of 
the  solstice,  in  June  or  December.  The  method  gives  a  fair 
approximation  at  any  time  of  year. 

12.  By  observation  of  the  North  Star  in  meridian. 

Find  the  time  of  passage  in  Table  I.  Choosing  still  weather, 
hang  a  plumb-bob  from  some  high  place  into  a  bucket  of 
water,  and  establish  a  point  of  observation  so  far  southward 
that  the  suspending  line  shall  cover  the  breadth  of  sky  be- 
tween the  Dipper  and  the  North  Star.  The  point  of  observa- 
tion may  be  an  upright  bodkin,  or  compass-sight,  fixed  on  a 
block  movable  horizontally  east  and  west.  Watch  for  the 
moment  when,  from  the  point  of  observation,  the  plumb-line 
covers  the  North  Star  and  the  first  star  in  the  handle  of  the 
Dipper;  that  is  to  say,  the  star  nearest  to  the  four  which 
make  the  Dipper  bowl.  Exactly  twenty-six  minutes  after- 
wards, bring  the  plumb-line  in  range  with  the  North  Star,  by 
shifting  the  observation  point  laterally.  That  range  -will  be 
the  true  meridian.  Stakes  may  be  set  in  it  forthwith  by 
means  of  candles. 

With  a  transit  in  good  adjustment,  the  plumb-line  is  not 
necessary.  Illuminate  the  cross  hairs  by  reflecting  light  on 
the  object  glass  from  white  paper. 

13.  Jjy  observation  of  the  North  Star  at  its  extreme  elonga- 
tion. 

Find  the  time  in  Table  II.,  and  make  the  preparations  above 
directed.  Keep  the  plumb-line  in  range  with  the  star  until 
the  star  apparently  ceases  to  move.  Mark  that  range.  Multi- 
ply the  natural  tangent  of  the  a/imuth,  given  in  Table  III.,  by 
the  distance  in  feet  from  the  point  of  observation  to  the  mark 
in  the  northern  range  just  set.  The  product  will  be  the  dis- 
tance from  said  northern  range  mark,  square  right  or  left,  to 
a  point  in  the  true  meridian  passing  through  the  point  of  ob- 
servation. If  the  western  elongation  was  observed,  set  off  the 
calculated  distance  eastward  from  the  northern  range  mark; 
if  the  eastern  elongation  was  observed,  set  the  distance  off 
westward.  If  both  the  eastern  and  western  elongations  be 


46  MISCELLANEOUS. 

observed,  the  true  meridian  will  pass  through  the  point  of 
observation,  bisecting  the  angle  between  the  northern  range 
marks. 

With  a  vernier  instrument,  the   azimuth  can  be  laid  off 
directly,  in  degrees  and  minutes. 


PROPOSITIONS   AND   PROBLEMS 
RELATING  TO  THE  CIRCLE. 

XVI.  -  XIX. 


PROPOSITIONS    AND    PROBLEMS 
RELATING  TO  THE  CIRCLE. 


XVI. 

PROPOSITIONS  RELATING  TO  THE  CIRCLE. 

The  following  propositions,  demonstrable  by  simple  processes 
of  geometrical  reasoning,  may  be  regarded  as  axiomatic. 


1.  In  any  circle  a  tangent  is  perpendicular  to  radius  at  the 
tangent  point.    Thus,  13 1  is  perpendicular  to  B  C. 

41) 


50  PROPOSITIONS   RELATING    TO    TUE   CIRCLE. 

2.  Tangents  drawn  to  a  circle  from  the  same  point  are  equal. 
Thus,  I  B  =  I E. 

.  3.  The  angle  DIE,  at  the  intersection  of  tangents,  is  equal 
to  the  central  angle  BO  E,  included  between  radii  to  the  tan- 
gent points. 

4.  If  a  chord  BE  connect  the  tangent  points,  the  angles 
I  B  E,  I  E  B,  are  equal :  each  of  them  is  equal  to  half  of  the 
central  angle  B C E,  or  of  the  intersection  angle  DIE. 

5.  Any  angle,  BCE,  at  the  centre,  subtended  by  the  chord 
BE,  is  double  the  angle  BFE,  at  the  circumference,  on  the 
same  side  of  the  chord  B  E. 

6.  Acute  angles  at  the  circumference,  subtended  by  equal 
chords,  are  equal. 

7.  An  acute  angle,  KFH,  between  a  tangent  and  a  chord, 
is  called  a  tanyential  angle,  and  is  equal  to  the  peripheral 
angle  LFH  subtended  by  an  equal  chord;  each  is  equal  to 
half  the  central  angles  FCH,  or  HCL,  subdivided   by  the 
same  chords. 

8.  The  exterior  angle  L  H  N  at  the  circumference,  between 
two  equal  chords,  is  called  a  deflection  angle :  it  is  equal  to  the 
central  angle,  or  to  twice  the  tangential  angle,  subtended  by 
either  chord. 

9.  U  F  K  be  made  equal  to  F  H,  and  H  N  be  made  equal  to 
HL,  H  K  is  called  the  tangential  distance,  and  LN  the  deflec- 
tion distance. 

10.  The  exterior  angle  E  HX  at  the  circumference,  between 
two  unequal  chords,  is  equal  to  the  sum  of  their  tangential 
angles,  or  to  half  the  sum  of  their  central  angles. 


XVII. 
CIRCULAR  CURVES  ON  RAILROADS. 

1.  The  circle  is  divided,  for  convenience,  into  360  equal 
parts,  called  degrees.  A  circle  36,000  feet  in  circumference 
would  be  cut  by  such  subdivision  into  360  parts,  each  100  feet 
long,  and  subtending  an  angle  of  one  degree  at  tin;  centre;  its 
radius  would  be  5,729.6  feet,  usually  reckoned  5,730  feet.  The 


CHICULAR    CUR  \'ES-  O2T  RAILKOADS.  SI 

chain  100  feet  long  being  the  unit  generally  adopted  by  Ameri- 
can engineers  for  field  measurements,  any  circular  arc  having 
that  radius,  of  5,7-30  feet,  is  called  a  one-degree  curve,  for  the 
reason  that  one  chain  is  equivalent  to  au  arc  of  one  degree  at 
the  circumference. 

2.  The  circumferences  of  circles  vary  directly  as  their  radii: 
hence,  in  any  circular  arc  struck  with  half  that  radius,  or 
2,865  feet,  one  hundred  feet  at  the  circumference  would  sub- 
tend an  angle  of  two  degrees  at  the  centre.    Such  an  arc  is 
called  a  two-degree  curve.     If  one-third  of  the  primary  radius 
of  5,730  feet,  or  1,910  feet,  be  used,  the  arc  is  called  a  three- 
degree  curve;  and  so  on. 

3.  It  should  be  borne  in  mind,  however,  that  these  measure- 
ments are  supposed  to  be  made  around  the  arc  itself,  and  not 
on  lines  of  chords.     Since  field  measurements  with  the  chain 
are  always  made  on  the  lines  of  the  chords,  which  are  shorter 
between  given  points  at  the  circumference  than  the  lines  of 
the  arcs,  as  a  bowstring  is  shorter  than  the  bow,  it  is  plain 
that,  in  advancing  towards  the  centre  of  the  one-degree  curve 
by  a  series  of  concentric  circles  having  radii  equal  to  one-half, 
one-third,  &c.,  of  the  primary  radius,  the  chord  100  feet  long 
d  iff  era  more  and  more  in  length  from  the  arc  subtended  by  it, 
the  bow  being  more  and  more  arched  in  relation  to  the  string. 
Thus,  in  the  circle  having  a  radius  equal  to  one-twentieth  of 
the  primary  radius,  the  chord  100  feet  long  subtends  an  angle 
of  20°  OG',  at  the  centre,  instead  of  20°,  and  the  arc  is  100.5 
feet  in  length,  instead  of  100  feet.     In  order,  therefore,  that 
the  chord  of  100  feet  may  subtend  arcs  of  1°,  2°,  3°,  &c.,  in 
regular  succession,  the  radii  of  these  successive  arcs  must  be 
somewhat  greater  than  the  above  method  by  subdivision  of  the 
primary  radius  would  make  them;  though,  as  might  be  inferred 
from  the  extreme  case  given  by  way  of  illustration,  the  dif- 
ference is  not  appreciable  in  ordinary  field  practice,  and  radii, 
together  with   all   the  functions    dependent    on    them,   may 
usually  be  held  to  vary  as  Hie. degree  of  curvature,  or  central 
angle  per  100  feet  chord,  varies. 


52 


TO  FIND    TDE  RADIUS   OF  A    CURVE. 


XVIII. 

TO  FIND  THE  RADIUS,  THE  APEX  DISTANCE,  THE 
LENGTH,   THE  DEGREE,   ETC.,   OF  A  CURVE. 

1.  Let  EB,  A  O  be  two  straight  lines  intersecting  at  E.     Lay 
off  equal  distances,  EA,  EB;  erect  perpendiculars  at  A  and 

B,  meeting  at  G,.and  con- 
n^ct  A  B,  E  G.  From  the 
centre  G,  with  radius  G  A, 
draw  the  curve  A  H  B. 

The  point  E  will  be  the 
P.  I. ;  A  and  B,  tangent 
points;  E  A,  E  B,  the  tan- 
gents, or  apex  distances, 
which  denote  by  T;  EH, 
the  external  secant,  or  S ; 
H  N,  the  versed  sine;  or 
V.  Let  the  long  chord 
AB,  connecting  the  tan- 
gent points,  be  called  C, 
Call  the  deflection  angle  to  a 


and  G  A  or  GB,  the  radius,  R. 

chord  of  100  feet  D,  as  before. 

2.  By  XVI.  3  and  4,  angle  E 


EBA=AGE 


o.    GIVEN  THE   INTERSECTION   ANGLE   I    AND   RADIUS  R,  TO 
9  FIND  THE  TANGENT  T. 

T  =  R  X  tan.  tr  I. 

Example. 
R  =  1,910.1,  I  =  35°  2-1'. 

Then  T  =  R  tan.  i  I  =  1,910.1  X  0.3191  =  C09.5. 

4.  Measure  from  the  P.  I.  equal  distances,  E  M,  E  F,  along 
the  tangents.  Measure,  also,  MF  and  EK,  the  distance  from 
E  to  the  middle  point  of  MF.  Then,  by  reason  of  similarity 
in  the  triangles  MEK,  EAG, 

M  K  :  E  K  :  :  A  G  :  A  E  :  :  R  :  T 

.-.  T  =  RXEK-j-MK. 


TO  FIND    THE  RADIUS   OF  A   CURVE.  53 

Example. 
Let  MK=  190.5,  EK  =  60.8,  R  =  1910.1. 

Then  R=  1910.1  ....  3.281056 
EK=  60.8  ....  1.783904 
MK  =  190.5  ( a.  c.)  .  .  .  7.720105 


T  =  609.6        ....        2.785065 

5.  If  100-feet  chords  be  used,  find  the  tangent  in  Table  XYI. 
corresponding  to  the  given  angle  I.  Divide  that  tabular  tan- 
gent by  the  degree  of  curvature  corresponding  to  the  given 
radius:  the  quotient  will  be  the  required  tangent.  Thus, 
Tab.  Tan.  corresponding  to  35°  24'  =  1,828.7,  which,  divided 
by  3,  the  degree  of  curvature,  gives  609.6,  the  tangent  sought. 


6.   GIVEN  THE   INTERSECTION  ANGLE  I   AND  TANGENT  T,   TO 
FIND   RADIUS   R. 

Transposing  the  equation  in  (3), 

R=T+  tan.  ±  I  =  T  X  cot.  £  I, 

Example. 
T=  609.6,  1=35°  24'  R  =  T  cot.  i  /  =  609.6  X  3.1334  =  1910.1. 

By  a  like  transposition  of  the  equation  in  (4), 


7.  If  100-feet  chords  be  used,  find  in  Table  XVI.  the  ^angent 
corresponding  to  the  given  angle  I.  Divide  that  tabular  tan- 
gent by  the  given  tangent;  the  quotient  will  be  the  degree  of 
curvature  in  degrees  and  decimals.  The  radius  corresponding 
to  this  degree  of  curvature  may  be  found  by  (12),  by  Table  X., 
or,  with  sufficient  accuracy  for  ordinary  practice,  by  dividing 
5,730,  the  radius  of  a  1°  curve,  by  it. 

Thus,  in  the  foregoing  example,  the  tabular  tangent  cor- 
responding to  35°  24'  is  1,828.7.  Dividing  by  609.6,  we  have  3 
for  the  degree  of  curvature;  and  5,730  divided  by  3  gives 
R  =  1,910  feet. 


54  TO  FIND    THE  RADIUS   OF  A    CURVE. 


8.    GIVEN   THE   INTERSECTION  ANGLE    I   AND  CHORD  A  B  —  C, 
CONNECTING   THE  TANGENT   POINTS,    TO   FIND   RADIUS   li. 

A  N  =  i  A  n  =  i  C;  A  G  N  =  i  I. 
AG  =  AN-^  sin.  AG  N  ; 
i.e.y  E  =  $  C -i- sin.  $  I. 

Example. 
I  =  35°  24',  (7=1161.4. 

Then  I?  =  -J  Gf  -=-  sin.  $!,  =  580.7  -7-  0.304  =  1910.2. 

9.  If  100-feet  chords  be  used,  find  in  Table  XVI.  the  chord 
corresponding  to  the  given  angle  I.  Divide  that  chord  by  the 
given  chord,  for  the  degree  of  curvature  in  degrees  and  deci- 
mals. Determine  the  corresponding  radius  by  (17),  by  Table 
X.,  or,  for  ordinary  practice,  by  dividing  5,730  by  it. 

Thus,  in  the  foregoing  example,  the  tabular  chord  corre- 
sponding to  angle  35°  24'  would  be  3,484.2,  which,  divided  by 
the  given  chord,  1,161.4,  gives  3  for  the  degree  of  curvature, 
arid  5,730  divided  by  3  makes  the  radius  R  =  1,910  feet. 


10.  GIVEN  THE  INTERSECTION  ANGLE  I  AND  THE  DEGREE 
OF  CURVATURE  OR  DEFLECTION  ANGLE  D,  WITH  100-FEET 
CHORDS,  TO  DETERMINE  THE  LENGTH  OF  THE  LONG  CHORD 
C,  THE  VERSED  SINE  Y,  THE  EXTERNAL  SECANT  S,  OK 
THE  TANGENT  T. 

Take  from  the  proper  column  in  Table  XVI.,  the  number 
corresponding  to  the  intersection  angle,  and  divide  it  by  the 
degree  of  curvature :  the  quotient  will  be  the  length  required. 

Example. 

A  4°  curve,  1  =  50°  16';  to  find  the  several  functions  above 
named. 

Table  XVI.  gives  the  designated  functions  of  a  1°  curve  as 
follows:  C  4,867.3,  V  542.4,  S  599.3,  T  2,688.2.  Dividing  by  4 
the  degree  of  curvature,  wre  have  for  the  corresponding  func- 
tions of  a  4°  curve  as  follows:  C  1,216.8,  V  135.6,  S  149.8,  T 
672.0. 


RADII,   DEFLECTION  ANGLES,    ETC.  55 


11.  GIVEN   C,    V,    S,    OR   T,  OF    ANY   CURVE,  AND  D,    THE   DE- 
GREE OF  CURVATURE,  TO  FIND  THE  INTERSECTION  ANGLE,  I. 

* 

Multiply  the  given  function  C,  V,  S,  or  T,  by  the  degree 
of  curvature,  D:  the  product  will  be  found  in  the  proper  col- 
umn of  Table  XVI.,  corresponding  to  the  required  angle. 

Example  1. 
Given  T  =  515,  D  =  5°;  to  find  I. 

Then  T  X  D  =  2,575,  which  corresponds  in  Table  XVI.  to 

48°  24'  =  I. 

Example  2. 

Given  C  =  1,656,  D  =  3°;  to  find  I. 

Then  C  X  D  =  4,968,  which  corresponds  in  Table  XVI.  to 
51°  23'  =  I. 

12.  GIVEN   C,  V,    S,    OR   T,    OF    ANY   CURVE,    AND  THE   INTER- 
SECTION ANGLE  I,  TO  P'IND  THE  DEGREE  OF  CURVATURE   D. 

Take  from  the  proper  column  of  Table  XVI.  the  number 
corresponding  to  the  given  an^le  I,  and  divide  that  tabular 
number  by  the  length  of  the  given  part;  the  quotient  will  be 
D,  in  degrees  and  decimals. 

Example  1. 

Given  T  =  587,  I  =  22°  26';  to  find  D. 
The  Tan.  corresponding  to  I  in  Table  XVI.  is  1,136.3. 

Then  1,136.3  -f-  587  =  1.935  =  1°  56'  =  D. 

Example  2. 

Given  S  =  64,  I  =  30°  25',  to  find  D. 
The  Ex.  Sec.  corresponding  to  I  in  Table  XVI.  is  208. 

Then  208  -=-  64  =  3.25  =  3°  15'  =  D. 


13.    GIVEN    THE    INTERSECTION    ANGLE    I,    AND    DEFLECTION 
ANGLE   D,   TO   FIND   THE   LENGTH  OF  THE   CURVE. 

Divide  I  by  D:  the  quotient  will  be  the  number  of  chord 
lengths  in  the  curve. 

If  the  degree  of  curvature  is  a  whole  number,  the  more  con- 
venient method  of  effecting  the  division  is,  first,  to  reduce  the 


RADII,   DEFLECTION  ANGLES,    ETC. 


minutes,  if  any,  in  I  to  decimals  of  a  degree;  then  divide  by 
the  degree  of  curvature. 

Example  1. 

I  =  20°  40',  I)  =  3°.  20°  40'  is  equivalent  to  20.67  degrees. 
Dividing  by  3,  we  have  6.80  chord  lengths  for  the  length  of  the 
curve.  If  the  chords,  as  is  usual,  are  each  100  feet  long,  the 
length  of  the  curve  in  this  case  will  be  689  feet.  If  the  chord 
lengthy  were  50  feet  eacii,  the  length  of  the  curve  would  be 
half  this  number  of  feet. 

14.  If  the  degree  of  curvature  is  fractional,  the  more  con- 
venient method  of  effecting  the  division  is,  first,  to  reduce 
both  1  and  D  to  minutes;  then  divide  the  former  by  the  latter. 

Example  2. 

I  ==  3()Q  22',  D  =  2°  45'.  These  are  equivalent,  respectively, 
to  1,822  and  165  minutes.  Dividing  the  former  by  the  latter, 
we  have  1,104  feet  for  the  length  of  the  curve. 

15.  The  ingenious   assistant  who  will   attentively   consider 
the  preceding  figures  cannot  fail  to  detect  other  obvious  analo- 
gies which  it  has  not  been  thought  necessary  to  include  in  this 
compendium. 

16.  In  railroad  field  practice  it  is  usually  sufficient  to  deter- 
mine angles  to  the  nearest  minute,  and  distances  to  the  nearest 
foot.     The  nicety  of  seconds  and  tenths  appears  generally  to 
be  quite  superfluous;  the  time  consumed  on  them  were  better 

employed  in  pushing  ahead. 


17.  GIVEN  ANY  DEFLECTION  AN- 
GLE D,  AND  CHORD  C,  TO  FIND 
RADIUS  R. 


FB  -I-  sin. 


L  B  =  B  L  ;  i.e., 


Example. 
Let  C=100  feet,  D  = 

Then  Z?  =  |  C  -f-  sin.  $ 
.0349  =  1432.7. 


=  50  -=- 


If  the  chords  are  100  feet  long,  as  is  usual  in  railroad  prac- 
tice, radius  may.be  found  with  sufficient  accuracy  by  dividing 


RADII,   DEFLECTION  ANGLES,  ETC.  57 

5,730,  Hie  radius  of  a  1°  curve,  by  the  deflection  Bangle,  or  de- 
gree of  curvature.  Thus,  in  the  foregoing  example,  5,730  -f-  4 
=  1,432.5. 

18.    GIVEN  ANY    RADIUS   R,  AND   CHORD   C,  TO   FIND  THE   DE- 
FLECTION  ANGLE  D. 

From  the  preceding  equation  and  example:  — 

Sin  f  D  =  i  C  -7-  K  =  50  -T- 1,432.7  =  .0349  —  xin  2°  =  £  D 


19.    GIVEN  ANY  RADIUS   R,  AND   CHORD   C,   TO   FIND   THE   DE- 
FLECTION DISTANCE    d. 

First  find  the  deflection  angle  by  above  method  (18).  Then, 
angle  HAB  in  the  figure  being  made  equal  to  D,  and  HA 
=  B  A,  B II  will  be  the  deflection  distance.  Draw  A  K  to  the 
middle  point  of  II  B. 

Then  d  =  HB  =  2  KB  =  2  AB  X  sin  KAB  =  2  C  X  sin 
|D. 

Example, 

Let  R  =  1,146  feet,  C  =  100  feet. 
By  (18)  D  will  be  "found  =  5°. 

Then  ( I  =  2  C  X  sin  I  D  =  200  X  .0436  =  8.72  feet. 

20.  If  the  chords  are  100  feet  long,  as  is  usual  in  field  meas- 
urement, divide  the  constant  number  10,000  by  the  radius  in 
feet:  the  quotient  will  be  the  deflection  distance.  The  deflec- 
tion distance  with  radius  of  10,000  feet  and  chord  of  100  feet 
is  one  foot:  this  rule  is  based  upon  the  principle  that  deflection 
distances;  the  chord  length  being  fixed,  will  vary  inversely  as 
the  radii. 

Thus,  in  the  foregoing  example,  10,000 -j-  1,140  =  8.72. 


21.    GIVEN  ANY  RADIUS  R,  AND  CHORD   C,  TO   FIND  THE  TAN- 
GENTIAL  ANGLE   T. 

The  angle  T  is  equal  to  \  U  by  construction;  for  mode  of 
determining  it,  see  preceding  section  (18). 


58 


ORDINATES. 


22.    GIVEN  ANY  RADIUS  R,  AND  CHORD  C,  TO  FIND  THE  TAN- 
GENTIAL  DISTANCE    I. 

First  find  the  tangential  angle,  as  above  directed.  Then, 
angle  13  A  E  in  the  figure  being  made  equal  to  T,  and  AE  = 
A  13,  BE  will  be  the  tangential-  distance.  Draw  AX  to  the 
middle  point  of  BE. 


Then 


EB  =  2  XB  =  2  AB  X  sin  N  AB  =  2  C  X  »m 


Example. 

Let  H  =  1,146  feet,  C  =  100  feet. 
By  sect.  1,  T  will  be  found  =  2°  30'. 

Then  t  =  2  C  X  sin  \  T  =  200  X  .0218  =  4.06  feet. 

23.  In  ordinary  railroad  practice  the  tangential  distance  may 
be  considered  equal  to  half  the  deflection  distance. 


XIX. 

OKDIXATES. 

1.    GIVEN   ANY   RADIUS   R,  AND   CHORD   C,   TO   FIND  THE   MID- 
DLE  ORDINATE  M. 


G- 'L 

Iii  the  annexed  figure,  II X  =  M,  H  G  =  R,  A  B  =  C. 


ORDINATKS.  59 


Example. 
=  819,  C  =100;  to  find  the  middle  ordinate,  M. 


M  =  819  —  V67076l^^2500  =  1.53. 


2.  Angle  HAN=MHGB;  HGB  =  £  AGB,  .'.  HAN  = 
±  A  G  B.  • 

UN  =  AN  X  tan.  HAN;  i.e.,  M  =  £  C  X  tan.  £  D;  D 
being  the  central  angle  subtended  by  the  chord. 

Example. 
D  =  7°,  C  =  100;  to  find  M,  the  middle  ordinate. 

M  =  i  C  X  tan.  £  D  =  50  X  0.03055  =  1.528. 

3.  GIVEN  THE  RADIUS  R,  CHORD  C,  AND  MIDDLE  ORDINATE 
M,  TO  FIND  ANY  OTHER  ORDINATE  E  K  =  M',  DISTANT  d 
FROM  N,  THE  MIDDLE  POINT  OF  THE  CHORD. 

KL  =  NG;NK=GL;EK  =  EL  — NG. 


E  L  =      G  E2  —  N  K2  =      R2  —  <i2;  N  G  (1) 

Then  E  K  =  M'  =  \/R2  —  &  —  VR2  —  ?  C*. 

4.  It  is  a  property  of  the  parabola,  that  ordinates  vary  as  the 
products  of  their  abscissas.     This  property  may  be  assigned  to 
the   circle   in   cases   where    the   arc  encloses   a  small  angle. 
Applying  it  here  we  have  — 

HN:EK::ANXNB:AKX  KB. 
Call  any  segments  AK,  K  13,  of  the  chord,  a  and  b. 
Then  M  :  M'  :  :  £  C2  :  ab,  .:  M'  =  M  X  4  ab  ^  C2. 

Example. 
M  =  1.528,  C  =  100,  a  =  60,  b  =  40;  to  find  M'. 

M'  =  1.528  X  9000  -i-  10000  =  1.528  X  0.96  =  1.467. 

5.  Multiply  the  corresponding  ordinate  of  a  1°  curve  from 
the  annexed  table  by  the  degree  of  curvature:  the  product  will 
be  the  ordinate  sought. 


no 


ORDINA  TES. 


ORDINATES  OF  A  1°  CURVE,  CHORD  100  PKKT. 


DISTANCES  OF  THE  ORDINATES  FROM  THE  END  OF  THE  IOO-FEET  CHOKD. 

Middle 

Feet. 

50 

Feet. 
45 

Feet. 
40 

Feet. 
•  35 

Feet. 
30 

Feet. 
25 

Feet^ 
20 

Feet. 
15 

Feet. 
10 

Feet. 
-      5 

LENGTHS  OF  THE  ORDINATES  IN  FEET. 

.218 

.216 

.209 

.198 

.183 

.164 

.140 

.111 

.078 

.041 

What  is  the  ordinate  of  a  G°  curve,  30  feet  from  the  end  oi 
the  100-feet  chord? 

The  corresponding  tabular  ordinate  of  a  1°  curve  is  .183 
which,  multiplied  by  6,  gives  1.098,  the  required  ordinate. 

(').  A  quick  way  of  laying  off  ordinates  on  the  ground,  and 
one  sufficiently  exact  for  the  field,  is,  after  fixing  the  point  II 
by  means  of  the  middle  ordinate  II X,  to  stretch  a  line  from 
II  to  A,  and  make  the  middle  ordinate  F  O  =  |  II X;  then  from 
F  to  A  and  F  to  II,  making  the  middle  ordinates  =  £  FO;  and 
so  on. 

7.  A  good  track-layer  will  seldom  require  points  at  shorten 
intervals  than  25  feet. 


TRACING  CURVES 

AND 

TURNING  OBSTACLES  IN  THE  FIELD. 
XX. -XXIII. 


TRACING-    CURVES  AND    TURNING 
OBSTACLES    IN  THE    FIELD. 


XX. 


TO  TRACE  A  CURVE  ON"  THE  GROUND  WITH  THE 
CHAIN  ONLY. 

1.  This  is  best  taught  by  an  example. 


Example. 

From  a  point  B,  18  feet  in  advance  of  A,  on  tangent  A  B,  to 
trace  a  curve  of  367  feet  radius  to  the  right,  with  chords  66  feet 
long,  and  consuming  an  angle  of  84°.  27'. 

63 


64  TO   TRACE  A    CURVE  ON  THE  GROUND. 

2.  First,  dividing  half   the  unit   chord,  or  33  feet,  by  the 
radius,  367  feet  (XVIII.,  18),  we  have  0.09+  for  the  sine  of  the 
tangential  angle,  corresponding  to  an  angle  of  5°  10':  the  de- 
flection angle,  therefore,  is  10°  20'.     The  tangential  distance 
corresponding  to  the  angle  5°  10',  and  chord  66  feet,  is  equal 
(XVIII.,  22)  to  twice  the  chord  multiplied  by  the  sine  of  half 
the  tangential  angle,  =  132  X  0.04507  =  5.95  feet.    The  deflec- 
tion distance  (XVIII.,  19)  is  equal  to  twice  the  chord  multi- 
plied by  the  sine  of  half  the  deflection  angle,  =  132  X  0.09+ 
=  11.88,  say  11.9  feet. 

3.  To  find  the  length  of  the  curve  (XVIII.,  13):  Divide  the 
total  central  angle  by  the  degree  of  curvature.     The  central 
angle,   34°  27',    is   equivalent   to  2067  minute's;    dividing  by 
620,  the  number  of  minutes  in  the  deflection  angle,  we  have 
3.33,  the  number  of  chord  lengths  in  the  curve,  =  3£  chains  = 
220  feet-. 

If  A  be  a  stake  numbered  2,  then  the  point  of  curvature,  B, 
will  be  2.18,  and  the  point  of  tangent,  F,  will  fall  at  2.18  + 
3.22  =  stake  5.40. 

4.  To  determine   the  tangential   distance  C  P,  to  the  first 
stake  on  the  curve,  either  of  two  methods  may  be  used:  — 

First,  The  sine  of  any  tangential  angle  is  equal  to  half  the 
chord  which  limits  the  angle  on  one  side  divided  by  radius. 
The  limiting  chord  B  C  in  this  instance  is  equal  to  66  —  18  = 
48  feet;  half  of  48,  therefore,  or  24  feet,  divided  by  radius,  367 
feet,  gives  0.0654,  the  sine  of  3°  45'  =  tangential  angle  PB  C. 
The  sine  of  half  this  angle  multiplied  by  twice  the  given  chord 
—  0.0327  X  96  =  3.14  feet,  the  tangential  distance  CP. 

5.  Secondly,  C  P  may  be  found  as  follows,  assuming  that 

the  functions  of  small 
angles  vary  directly  as 
the  angles  themselves, 
and  vice  versa. 

Let  BF  be  a  portion 
of  the  curve.  Make  the 
tangent  B  E  equal  to  the 
chord  B  F,  and  strike  the 
arc  E  F.  Draw  the  sub- 
chord  B  C,  and  strike  the 

arc  C  P.  Prolong  B  C  to  D.  E  F  may  be  taken  as  the  tangen- 
tial distance  due  to  the  whole  chord  BF,  and  PC  the  tangen- 
tial distance  due  to  the  sub-chord  B  C. 


TO    TRACE  A    CURVE  ON  THE  GROUND.  55 

Then  PC  :  E  D  ::  B  C  :  BD  or  BF;  and,  by  the  foregoing 
supposition,  E  D  :  E  F  : :  B  C  :  B  F.  Combining  these  propor- 
tions, and  cancelling  E  D,  we  have  P  C  :  E  F  : :  B  C2  :  B  F2  .  \ 
PC  =  EF  X  (BC-f-BF)2. 

In  words,  the  tangential  distance  for  a  sub-chord  is  to  that 
for  a  whole  chord  as  the  square  of  the  sub-chord  is  to  the 
square  of  the  whole  chord.  The  same  is  true  of  deflection  dis- 
tances. 

6.  In  the  example  we  are  treating,  the  tangential  distance  for 
the  whole   chord   of  66  feet  has  been  found  to  be  5.95  feet; 
that  for  48  feet,  therefore,  is  5.95  X  48-  -h  662  =  5.95  X  0.528 
=  3.14,  as  before. 

Stretch  the  48  feet  of  chain  from  B  to  P,  in  prolongation  of 
tangent  A  B,  and  mark  the  point  P ;  then  step  aside,  and  stretch 
from  B  to  C,  making  the  distance  PC  =  3.14  feet:  C  will  be  a 
stake  on  the  curve. 

7.  Next,  run  out  the  whole  chain  length  from  C  to  O  in  the 
range  B  C.     To  find  O  D,  suppose  the  line  X  C  T  to  be  drawn 
tangent  to  the  curve  at  C.     Then  X  D  may  be  considered  the 
tangential  distance  due  to  the  whole  chord,  =  5.95,  as  above 
determined. 

The  angle  O  C  X  =  T  C  B  =P  B  C  (XVI.,  4) ;  and  (5) 

OX:XD::B.C:CD.\  OX  =  XDXBC-=-CD;  i.e.,  OD 
=  X  D  +  0  X  =  X  D+  [X  D  X  (B  C^C  D)]  =5.95  X  [1  +  (48 
4-06)]  =5.95  X  1.727  =  10.27. 

8.  The  point  X  may  be  fixed  otherwise  by  laying  off  B  T  = 
CP,  and  running  out  the  chain  length  CX  in  the  range  C  T. 
The  point  D  on  the  curve  may  then  be  fixed  by  making  XD 
equal  to  5.95  feet,  the  tangential  distance. 

Xext  run  out  the  chain  to  M,  in  the  range  CD;  make  ME 
equal  to  the  deflection  distance,  11.9  feet,  and  fix  the  point  E. 
The  points  C,  D,  and  E  will  be  stakes  3,  4,  and  5  on  the  curve. 

9.  To  set  the  point  of  tangent,  F,  at  stake  5.40,  prolong  the 
chord  line  D  E  for  40  feet  to  L,  and  suppose  V  E  to  be  drawn 
tangent  to  the  curve  at  E.     Then  the  angle  L  E  V  is  equal  to 
the  tangential  angle  of .  the  curve ;  and  the  sub-tangential  dis- 
tance L  V  is  to  the  whole  tangential  distance  due  to  the  66- 
feet  chord,  as  the  sub-chord  is  to  the  whole  chord  (5);  i.e., 
L  V  =  5.95  X  40  -r-  66  =  3.6  feet. 

By  the  method  illustrated  in  (6),  the  distance  FV  will  be 


66  TO   TRACE  A    CURVE  ON  THE  GROUND. 

* 

equal  to  5.95  X  402  -f-  662  =  5.95  X  0.367  =  2.18  feet.  With 
the  distance  LF  =  3.6  -j-  2.18  =  5.78  feet,  thus  obtained,  and 
the  sub-chord  E  F  =  40  feet,  the  point  of  tangent  F  may  be 
established. 

10.  Next,  set  off  U  E  =  F  V  =  2.18  feet,  and  lay  out  FK  in 
prolongation  of  the  range  UF;  FK  will  be  in  the  line  of  the 
terminal  tangent. 

11.  This  analysis  has  been  somewhat  minute  and  detailed, 
in  order  that  the  subject  may  be  thoroughly  understood.     An 
instrument  for  measuring  angles  should  always  be  used  in  rail- 
road service:   it  greatly  simplifies  and   abridges  the  labor  of 
tracing  field-curves,  and  gives  more  exact  results.     But  occa- 
sions sometimes  rise,  in  miscellaneous  practice,  when  strict 
accuracy  is  not  required,  and  the  chain  only  can  be  had:  the 
young  engineer  should  qualify  against  such  occasions. 


XXI. 

TO  TRACE   A   CURVE   ON  THE   GROUND  WITH 
TRANSIT  AND   100-FEET   CHAJN. 

1.  This,  also,  is  best  taught  by  an  example. 

Let  it  be  a  general  rule,  in  locating,  to  fix  the  intersection  of 
tangents,  and  to  set  the  tangent  points,  or  the  P.  C.  at  least, 
from  the  P.  I.  There  are  exceptional  conditions,  as  a  steep 
hillside,  timber  or  broken  ground,  a  very  long  arc,  unimpor- 
tance of  exact  conformity  to  the  project,  and  the  like,  which 
warrant  its  omission;  but  where  these  conditions  do  not  obtain 
or  are  not  prohibitory,  and  a  snug  fit  is  desirable,  time  will 
usually  be  saved  by  fixing  the  P.  I.  It  often  proves  serviceable 
as  a  reference  point  during  construction:  on  the  location,  it 
gives  confidence  in  the  work  and  an  assurance  of  safe  progress, 
which  are  well  worth  a  little  painstaking  beforehand. 

2.  Having  established  the  P.  I.,  and  found  the  intersection 
angle  to  measure,  say,  66°  45r,  the  first  step  is  to  find  the  apex 
distances  so  called,  or  tangent  lengths  IB,  IF.     These  are 
each  equal  to  R  X  tan.  1 1.     If  a  7°  30'  curve  be  prescribed  to 
close  the  angle,  R  X  tan.  1 1  =  764  X  0.659  =  503  feet, 


TO    TRACE  A    CURVE  ON  THE  GROUND. 


67 


Or,  referring  to  Ta- 
ble XVI.,  the  tangent 
corresponding  to  66° 
45'  is  found  by  inter- 
polation to  be  3774.6; 
dividing  by  7.5,  the 
rate  of  curvature  in 
degrees  and  decimals, 
we  have  for  the  apex 
distance  503  feet,  as 
above. 

3.  Before  disturbing 
the  instrument,  which 
is  presumed  to  stand 
in  the   range    of    the 
terminal     tangent, 
measure    I  F,  =  503 
feet,  and  set  the  P.  T. 
at  F.     Then  direct  the 
telescope   to   the    last 
point  fixed  on  the  ini-  \ 
tial  range  AB,  meas- 
ure I  B,  =  503  feet, 
and  set  the  P.  C.  at  B. 
Move  to  B. 

4.  Suppose  the  P.  C. 
to    have    fallen    at   a 
stake  2.50.     In   order 
to  find  the  length  of 
the  curve,  divide  the 
intersection  angle  by 
the   degree   of  curva- 
ture, having  first  re- 
duced the  minutes  in 
eachtohundreths  of  a 
degree  by  multiplying 
by  10  and  dividing  the 
product   by   6.     Thus 
the  intersection  angle 
becomes    66.75°,    and 
the   degree   of  curva- 
ture 7.5°:  dividing  the 


68  TO    TRACE  A    CURVE   ON   THE   GROUND. 

former  by  the  latter,  we  have  890  feet  for  the  length  of  the 
curve. 

Or,  the  intersection  angle  66°  45'  is  equivalent  to  4005',  and 
the  degree  of  curvature  7°  3D'  is  equivalent  to  450':  dividing 
the  former  by  the  latter,  we  have  890  feet  for  the  length  of  the 
curve,  as  before. 

5.  Adding  8.90  to  2.50,  the  number  of  the  P.  C.,  the  P.  T.  is 
found  to  fall  at  stake  11.40.     Let  the  rear  chainman  make  a 
note  of  this,  that  there  may  be  no  mistake  in  the  terminal  plus. 

6.  Next,  to  determine  the  proper  deflections  from  the  line  oi 
tangent  at  B,  bear  in  mind   that   the   deflection   for  a  whole 
chain  is  half  the  degree  of  curvature ;  and  that,  in  field-curves 
of  more  than  300  feet  radius,  the  deflections  for  sub-chords, 
or  plusses,  may,  without  material  error,  beheld  to  vary  directly 
as  the  sub-chords  themselves;  that  is  to   say,  the  sub-deflec- 
tions due  to  30,  60,  and  80  feet,  for  instance,  will  be,  to  the 
deflection  due  to  100  feet,  as  30,  60,  and  80  are  to  100. 

7.  Thus,  in  the  example,  7°  30'  being  the  degree  of  curva- 
ture, one-half  of  this,  or  3°  45',  will  be  the  deflection  due  to  a 
chord   of   100  feet;  and  ^  of  this,  or  a  deflection  of  1°  52| 
from  the  line  of  tangent  at  B,  will  fix  stake  3,  50  feet  distant 
on  the  curve. 

8.  The  following  is   a  simple  rule   for  finding  sub-deflec- 
tions :  — 

Multiply  the  sub-chord  in  feet  by  the  rale  of  curvature  in 
degrees  and  decimals:  three-tenths  of  the  product  will  be  tht 
sub-deflection  in  minutes. 

Thus,  in  the  example,  50  X  7.5  =  375,  and  375  X  0.3  = 
'112.5'  =  1°  521',  as  before. 

9.  Having  set  stake  3,  stakes  4  and  5  will  be  fixed  by  succes- 
sive deflections  of  3°  45'.     In  establishing  stake  5,  the  index 
will  read,  1°  52|'  +  3°  45'  +  3°  45'  =  9°  22f  =  angle  CBS. 

10.  Suppose  the  instrument  moved  to  5.     See  that  the  ver- 
nier has  not  been  disturbed,  backsight  to  B,  and  deflect  9°  22£ 
right;  i.e.,  double  the  index  angle.     The  index  will  now  read, 
18°  45'  =  the  angle  I  CD;  and  the  telescope  will  be  directed 
along  the  line  C  D,  tangent  to  the  curve  at  5,  for  the  reason 
that  the  angle  B5C  has  been  made  equal  to  the  angle  CB£ 
(XVI.  4). 

Proceed  with  successive  deflections  of  3°  45'  from  this  tan- 
gent, and  set  stakes  6,  7,  8,  and  9,  at  intervals  of  100  feet, 

11.  Suppose  the  instrument  moved  to  9.      In  fixing  this 


TO   TRACE  A    CURVE  ON   THE   GROUND.  69 

stake,  the  index  will  read,  18°  45'  +  4  times  the  constant  angle 
30  45/}  =  igo  45'  _|_  150  _  ailgie  I  c  D  +  angle  D  5  9,  =  33° 
45'.  In  order  to  place  the  telescope  in  the  line  D  E,  tangent  to 
the  curve  at  9,  it  is  now  necessary  to  turn  an  angle  to  the 
right,  from  backsight  to  5,  equal  to  D95  =  D59  =  15°;  i.e., 
the  vernier  should  be  moved  from  33°  45'  to  33°  45'  -f-  15°  = 
48°  45'.  The  telescope  will  then  be  in  tangent  at  9. 

12.  A  simple  rule  for  finding  the  index  angle  which  shall 
place  the  telescope  in  tangent  at  any  point  on  the  curve  is  as 
follows :  — 

From  double  the  index  angle  which  fixed  the  given  point,  sub- 
tract the  index  reading  in  tangent  at  the  last  turning-point :  the 
remainder  will  be  the  required  index  angle. 

Thus  the  index  angle  which  established  stake  9  was  33° 
45'.  Double  this  angle  will  be  67°  30' ;  subtracting  18°  45',  the 
reading  in  tangent  at  the  last  turning-point,  we  have  48°  45', 
the  required  index  angle,  as  before. 

The  reasons  for  the  rule  will  be  obvious  from  an  examina- 
tion of  the  figure. 

13.  Being  in  tangent,  then,  at  9,  and  the  index  reading  48° 
45',  a  deflection  of  3°  45'  will  fix  10:  a  further  deflection  of  3° 
45'  will  fix  11,  and  the  index  will  stand  at  48°  45'  +  7°  30'  = 
56°  15'. 

14.  To  find  the  deflection  corresponding  to  the  sub-chord  11 
F,  =  40  feet :  by  the  foregoing  rule  (8),  the  degree  of  curva- 
ture, 7.5,  multiplied  by  40,  the  length  of  the  sub-chord  in  feet, 
gives  a  product  of  300,  three-tenths  of  which  amount  to  90 
minutes  .=  1°  30'.     Adding  1°  30'  to  56°  15',  makes  the  index 
angle  57°  45'  to  fix  the  P.  T.  at  11.40. 

15.  Move  to  the  P.  T.  at  11.40,  see  that  the  vernier  has  not 
been  disturbed,  and   backsight  to  9.     By  the  foregoing  rule 
(12),  double  the  index  angle,  57°  45',  less  the  angle  in  tangent 
at  9,  the  last  turning-point,  48°  45',  =  115°  30'  —  48°  45',  = 
66°  45',  =  the  index  angle  in  tangent  at  the  P.  T.,  =  the  total 
angle  consumed  by  the  curve.     The  work  thus  proves  itself. 

16.  The  preceding  example  would  appear  in  the  field-book 
as  follows:  — 


TO    TRACE  A    CURVE  ON  THE  GROUND. 


•&     « 

a    s 

1  ? 


.   . 
•<     < 


^      <>1       O       iO       O 

OOlCOr-lO 


?H      °M      ?0 


0  ©  00 


ri    f^    -r 


<»    «     o 


TURNING   OBSTACLES   TO   VISION  IN  TANGENT.  71 

17.  This  mode  of  running  curves  secures  a  record  of  each 
step  in  the  proceeding;  so  that,  if  any  error  occurs,  it  can 
readily  be   detected.     At  each  turning-point,  the  number  in 
the  "  tangent"  column  must  correspond  with  the  central  angle 
due  to  the  length  of  curve  to  that  point;  and  at  the  P.  T.  that 
number  must  correspond  with  the  total  central   angle.     The 
work  can  thus  be  checked  with  facility  during  its  progress, 
and  checks  itself  at  the  end. 

18.  The  young  transitman  is  recommended  to  rule  blanks 
after  the  pattern  given,  and  exercise  himself  thoroughly  in 
computing  the  parts,  and  recording  the  field-notes  of  various 
curves  assumed  at  will:  drawings  are  not  necessary. 


XXII. 

TURNING   OBSTACLES   TO   VISION    IN  TANGENT. 

1.  Draw  CF  parallel  to  AB.     Let  lines  BC,  CE,  FG,  cut 

these  parallels  at  equal 
inclinations.   Call  this 
angle  I.    Then  B  C  = 
CE  =  FG.      BE  =    A— B- 
BD  +  DE  =  2BD. 

But  BD  =  BC  cos.  I,  .-.  BE  =  2  BC  cos.  I.     EG  =  C  F. 

cos.  I. 


Example. 

Suppose  B  to  be  a  stake  24.50  on  the  tangent  A  B,  and  that 
a  deflection  left  of  10°  be  made  there  for  200  feet  to  a  point  C. 
Set  transit  at  C,  vernier  reading  10°  left.  B  S  to  B,  and  deflect 
20°  right.  Vernier  will  now  read  10°  right,  and  telescope  will 
be  in  line  C  E.  'Hake  C  E  =  200  feet.  Move  to  E.  See  that 
vernier  still  reads  10°  right.  B  S  to  C,  arid  turn  10°  left.  Ver- 
nier will  now  read  zero,  and  telescope  will  be  in  line  E  G,  or 
tangent  A  B  prolonged. 

Distance  BE  =  2  B  C  cos.  1  =  2  (200  cos.  10°)  =400  X  .985 
=  394  feet.  Then  E  =  24.50  +  394,  =  stake  28.44  on  tangent 
A  B  prolonged. 


72  TURNING   OBSTACLES   TO    VISION  IN  TANGENT. 

If  a  parallel  line  C  F  were  run,  a  deflection  of  10°  right  would 
be  made  at  each  of  the  points  C  and  F.  If  C  F  were  250  feet, 
then  B  G  would  be  =  250  +  394  ==  644  feet,  and  the  point  G 
would  fall  at  stake  30.94  on  tangent  A  B  prolonged. 

2.  If  angle  I  ==  60°,  the  other  conditions  of  above  method 
being  observed,  triangle  BHE  will  be  equilateral,  and  BE  = 

B  H  =  H  E.  If  the  parallel  D  C 
or  DF  be  run,  BE  =  BD  -f 
DC,  and  BG  =  BD  +  DF. 
For  field  work  see  last  example. 
3.  In  turning  obstacles  by 
either  of  these  methods,  the 
angles  should,  be  measured  with  extreme  niceness.  Handle 
the  instrument  lightly,  to  avoid  jarring  the  vernier;  and,  if 
possible,  observe  well-defined  distant  objects  in  the  several 
short  ranges,  that  the  lines  of  foresight  and  backsight  may 
accurately  coincide. 

In  locating,  the  following  method  is  preferable  to  those  given 
above,  and  should  always  be  used  on  long  tangents. 

4.  Having  established  points  A  and  B  on  the  centre  line, 
the  farther  apart  the  better  within  limits  of  distinct  vision, 
set  off  the  equal  £  F  G  H 

rectangular    d  i  s  - 

tances    A  E,    B  F,  I    / 

„    — i LJj 

ranging     clear    of      A  Bjf 

the      obstacle. 

Place  the  transit  at  E  or  F,  fix  points  G  and  H  on  the  forward 
range,  and,  rectangularly  to  these  points,  establish  others  on 
the  forward  range  of  the  centre  line  at  C  and  D.  The  offset 
distances  should  be  measured  very  carefully  with  the  rod,  or 
with  a  steel  tape  if  they  exceed  in  length  the  pocket  rule 
which  every  engineer  should  have  about  him. 


TURNING   OBSTACLES   TO  MEASUREMENT  IN  TANGENT.  73 


XXIII. 

TURNING  OBSTACLES  TO  MEASUREMENT  IN 
TANGENT. 

1.  Fix  a  point  on  tangent  A  B  prolonged  at  E.     Lay  off  at  B 
a  perpendicular  of  any  convenient  length.     Move  the  instru- 
ment to  D,  make  the  angle  B  D  A 

=  B  D  E,  and  mark  the  point  of 
intersection  A.  By  reason  of 
symmetry  in  the  triangles  A  D  B, 
BDE,  AB  =  BE,  and  may  be 
measured  on  the  ground. 

2.  Or,  fix  the  point  E,  arid  lay 

off  the  perpendicular  BD  as  before.  Move  to  D,  direct  the 
telescope  to  E,  turn  a  right  angle  E  D  C,  mark  the  point  of 
intersection  C,  and  measure  C  B.  Then,  by  reason  of  simi- 
larity in  the  triangles  C  B  D,  D  B  E,  C  B  :  B  D  :  :  B  D  :  B  E, 


Example. 

Suppose  B  D  to  be  60  feet,  and  B  C  40  feet.     Then  B  D  2  -h 
B  C  =  3600  -+-  40  =  90  feet  =  BE. 

3.  Or,  with  the  instrument  at  D,  measure  the  angle  BDE. 
Then  B  E  =  B  D  tan.  BDE. 

Example. 

B  D  =  120  feet,  angle  B  D  E  =  54°  40'.     B  D  tan.  B  D  E  = 
120  X  1.41  =  169.2  feet  =  B  E. 

4.  Or,  without  an  instrument,  lay  off  any  convenient  lines  B  F 

or  C  H.  Mark  the  middle  point 
D.  Line  out  H  G,  parallel  to 
AB.  Mark  on  it  the  point  G 
in  range  with  D  and  E.  Then 
GF  =  BE,  orGH  =  CE. 

5.  Should  the  use  of  a  right 
angle  be  inconvenient,  turn  any 
angle  E  B  D  =  x,  measure  B  D 
about  equal  by  estimation  to  B  E,  if  the  ground  permits,  and 
set  a  point  D.  Move  to  D,  and  measure  angle  15  D  E  =>  y. 


74    TURNING   OBSTACLES    TO  MEASUREMENT  IN  TANGENT. 

Then  the  angle  BED,  or  z,  — 180  —  (cc  -}-?/),  and,  by  trigonom- 
etry, sin.  z  :  sin.  y  \\  BD:BE, .'.  B  E  =  B  D  sin.  y  4-  sin.  z. 

Example. 
Let  x  =  44°  02',  y  =  71°  48',  B  D  =  300  feet. 

Then  z  =  180°  —  (x  +  y)  =  180°  —  (44°  02'  -f  71°  48')  = 
180°  —  115°  50;  =  64°  10'.  B  E  =  B  D  sin.  y  4-  sin.  z  =  300 
sin.  71°  48'  ~  sin.  64°  10'  =  300  X  .95  -f-  .90  =  316.6  feet. 

The  calculation  by  logarithms  would  be  as  follows:  — 

Log.  300 2.477121 

9.977711 


Sum 12.454832 

Log.  sin.  64°  10' 9.954274 

Log.  316.6.     Diff.    . 2.500558 

If  E  is  invisible  from  B,  extend  the  line  D  B  towards  C, 
until  a  line  C  E  clears  the  obstacle.  The  point  E  must  then 
be  established  by  intersection  of  the  sides  CE,  DE,  in  triangle 
C  D  E.  Supposing  the  extension  B  C  to  have  been  120  feet, 
the  side  CD  will  be  420  feet,  the  angle  y  71°  48';  and,  by  a 
calculation  similar  to  the  above,  the  side  DE,  opposite  angle  x 
in  the  lesser  triangle,  identical  with  DE  in  the  larger  one,  will 
be  found  to  be  231.7  feet.  The  sum  of  the  angles  at  the  base 
C  E  of  the  triangle  C  D  E  =  180°  —  y  =  180°  —  71°  48'  =  108° 
12'.  By  trigonometry,  two  sides  and  the  included  angle  being 
known  in  any  plane  triangle,  the  sum  of  the  known  sides  is  to 
their  difference  as  the  tangent  of  the  half  sum  of  angles  at  base 
is  to  the  tangent  of  half  their  difference.  In  triangle  ODE, 
therefore,  CD  +  DE  or  651.7  :  CD  — DE  or  188.3  ::  tan. 
108.12  -t-  2  or  tan.  54°  06'  :  tan.  54°  06'  X  188.3  -^  651.7  = 
.399  =  tan.  21°  45',  =  half  the  difference  of  the  angles  at  the 
base. 

Log.  188.3 2.274850 

Tan.  54°  06' 0.140334 

Sum 2.415184 

Log.  651.7 2.814048 

Tair?21°45',     Diff..  ,     9.601136 


TURNING   OBSTACLES   TO  MEASUREMENT  IN  TANGENT.  75 

The  angle  at  C,  being  evidently  the  lesser  of  the  two  angles 
at  the  base,  is  equal  to  the  half  sum  of  these  angles  decreased 
by  their  half  difference,  =  54°  06'  —  21°  45'  =  32°  2V. 

Set  the  transit,  then,  at  C,  foresight  to  D,  deflect  32°  21'  left, 
and  fix  in  that  range  two  points  F  and  G,  between  which  a 
cord  may  be  stretched,  and  as  nearly  as  can  be  judged  on 
opposite  sides  of  E.  Move  to  D,  foresight  to  C,  deflect  71°  48' 
right,  and  establish  a  point  E  at  intersection  with  FG.  Cross 
to  E,  B  S  to  D,  and  deflect  the  angle  z  =  64°  KX  into  the  line 
of  the  tangent  A  B  prolonged. 


SUGGESTIONS  AS  TO   FIELD-WORK 
AND   LOCATION -PROJECTS. 

XXIV. -XXV. 


SUGGESTIONS  AS  TO  FIELD-WORK 
AND  LOCATION -PROJECTS. 


XXIV. 

SUGGESTIONS  CONCERNING  FIELD-WORK. 

1.  THE  CHIEF  ENGINEER,   after  conference  with  his  em- 
ployers in  regard  to  the  character  of  the  work  contemplated, 
and  its  general  route,  should,  before  organizing  field-corps,  go 
over  the  ground  in  hoth  directions,  and,  aided  by  the  best 
attainable  maps,  qualify  himself  by  actual  observation  to  in- 
struct his  assistants  as  to  the  conduct  of  the  survey.    Equipped 
with  hand-level,  pocket-compass,  and  in  rough  regions  with 
the  aneroid,  he  can  often  not  only  prescribe  lines  for  examina- 
tion, but  indicate  the  gradients  to  be  tried,  thus  saving  a  vast 
amount  of  random  labor  and  needless  expense.     Such  thorough 
preliminary  exploration  is  due  both  to  himself  and  his  princi- 
pals: it  is  too  often  omitted,  or  done  with  a  perf unctory  rush. 
In  broken  topography,  no  maps,  notes,  or  information  derived 
from  others  can  supply  the  want  of  personal  acquaintance  with 
the  ground  itself.     He  must  indispensably  make  that  acquaint- 
ance, in  order  to  project  an  intelligent  location,  —  a  work  which 
should   rarely  be  delegated;  being  capital   service,   it   comes 
within  the  special  function  of  the  chief  engineer,  and  oily  the 
necessary  distribution  of  labor  attending  a  great  charge  should 
relieve  him  from  its  direct  performance. 

2.  A  FIELD-CORPS  in  settled  regions  generally  consists  of 
one  senior  assistant  or  chief  of  corps,  one  transitman,  one 
leveller,  one  rodman,  two  chainmen,  one  slopeman,  and  two 
or  more  axemen. 

The  following  notes  in  regard  to  the  allotment  of  duties  and 
the  conduct  of  work  may  be  acceptable.  They  are  copied 
from  the  writer's  memoranda  for  the  guidance  of  his  field- 
parties,  with  the  addition  of  some  detail,  and  practical  hints 
here  and  there,  to  aid  the  inexperienced. 

79 


80  SUGGESTIONS   CONCERNING  FIELD-WORK. 

3.  THE  SENIOR  ASSISTANT  will  receive  instructions  from 
the  principal  assistant  in  charge,  or  the  chief  engineer,  and 
will  act  exclusively  under  his  direction. 

He  will  be  held  responsible  for  the  good  conduct  of  the  corps, 
and  for  the  rapid,  exact,  and  economical  performance  of  the 
work.  Indecent  or  blasphemous  outcries  in  the  field  should 
be  prohibited.  The  writer's  various  travel  by  land  and  sea 
has  brought  him  acquainted  with  many  cultivated,  estimable, 
energetic,  profane  fellows,  but  not  one  in  whom  swearing  was 
a  grace;  nor  has  he  ever  seen  an  instance  where  it  forwarded 
work.  Those  considerate  of  others'  pride  and  self-respect 
will  generally  find  that  a  good  leader  makes  good  followers. 

The  senior  assistant  is  empowered  to  appoint  and  dismiss 
employes  below  the  rank  of  rodman,  and  will  report  any 
inefficiency  or  neglect  of  duty  in  the  ranks  above  to  his 
chief. 

He  will  pay  the  authorized  expenses  of  the  corps  for  sup- 
plies, repairs,  transportation,  and  subsistence,  taking  duplicate 
vouchers.  Accommodations  should  be  sought  near  the  work. 
When  not  thus  obtainable,  transportation  to  and  from  the 
field  is  to  be  regarded  as  a  measure  of  economy  for  the  com- 
pany, compensating  the  expense  incurred  by  saving  time  and 
labor. 

He  will  superintend  field  operations  in  person,  keeping  in 
advance  of  the  transit  to  direct  and  expedite  the  work,  and 
establish  the  turning-points.  On  preliminary  surveys,  the  axe 
should  be  little  used ;  and  on  alternative  locations,  or  such  as 
may  be  subject  to  revision,  trees  over  four  inches  in  diameter 
need  rarely  be  felled. 

He  should  be  patient  with  sensitive  landholders.  He  will 
find  exercise  for  that  amiable  virtue,  also,  with  the  field  vis- 
itors who  so  often  spare  time  from  useful  toil  to  tell  him  he  is 
on  the  wrong  line,  and  to  show  him  where  the  right  one  is. 

Note  for  record  the  kind  and  quality  of  material  to  be  moved, 
observing  quarries,  wells,  or -other  indications  for  the  purpose; 
the  timber  and  rock  in  the  country  traversed,  with  a  view  to 
their  use  in  construction,  and  the  widths  of  passage  to  be  pro- 
vided for  streams,  together  with  the  character  of  their  banks 
and  beds. 

Note  the  names  of  residents  in  the  immediate  vicinity  of  the 
work  on  survey;  and,  on  location,  cause  the  property-lines  to 
be  observed  and  recorded  also  when  convenient. 


SUGGES  TIONS    CONCERNING  FJEL D  -  WORK. 


81 


Always  begin  grade-lines  at  the  summit,  and  work  down. 
For  such  service,  carry  habitually  a  slip  of  profile  paper,  say 
six  inches  wide  and  two  feet  long.  Rule  the  proposed  grade- 
line  on  it,  assume  a  summit  cut,  mark  the  stations,  and  start 
down.  When  at  fault,  the  elevation  can  be  spotted  on  the 
profile,  which  will  show  at  a  glance,  without  any  calculation, 
how  you  stand  in  relation  to  grade. 

The  work  of  each  day  should  be  compiled  and  recorded  in 
the  evening,  that  no  delay  may  result  from  the  loss  or  deface- 
ment of  a  field-book. 

FORM    FOB   SURVEY   RECORD. 


STA. 

Dis. 

DEFLEC. 

COURSE 

M.C. 

ELEVA. 

SLOPE. 

REMARKS. 

FORM   FOR   LOCATION  RECORD. 


w     i 

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• 

h£ 

B 

"         2 

Z 

0 

STATION 

DISTANC 

COURSE. 

a   i  w 

|  GRADIE 

GRADE. 

VARIATI 

SLOPE. 

REMARKS. 

i 

On  location,  check  the  transitman's  calculation  of  the  length 
of  each  curve  and  the  fractional  deflections. 

The  senior  assistant  must  be  qualified  to  locate  a  line  accu- 
rately on  the  ground  from  the  project  furnished  him.  Lateral 
deviations  exceeding  five  feet  on  ten-degree  slopes,  three  feet 
on  fifteen-degree  slopes,  and  two  feet  on  twenty-degree  slopes, 
will  be  considered  errors  requiring  correction.  Measurements 
to  the  experimental  line  should  be  made  and  noted  frequently, 
in  order  not  only  to  check  the  field-work,  but  that  the  line 
may  by  means  of  them  be  laid  down  on  the  map. 

The  senior  assistant  will  supply  himself  with  drawing  in- 
struments, colors,  brushes,  and  the  like  personal  furniture  of 
an  engineer.  He  will  take  care  also  that  the  stationery,  field- 
books,  instruments,  and  other  articles  of  outfit  supplied  by  the 
company,  are  not  misused.  His  field  equipment  should  always 
include  a  hand-level  and  a  pocket-compass:  to  these  may  be 


82  SUGGESTIONS   CONCERNING  FIELD-WORK. 

added  a  straight,  round  staff,  five  or  six  feet  long,  steel  pointed; 
it  will  be  found  exceedingly  useful. 

If  without  a  topographer,  he  should  make  sketches  of  irregu- 
lar ground,  of  streams,  buildings,  roads,  and  the  like,  to  help 
in  compiling  the  map. 

In  hilly  or  wooded  districts,  the  front  chainmaii  carries  the 
flag  on  survey,  and  is  at  the  head  of  the  line.  In  open,  plain 
country,  work  is  greatly  forwarded  by  detaching  an  axeman 
with  flag,  to  accompany  the  senior  in  advance,  and  set  turning- 
points  for  the  transit.  The  transitmaii  follows  as  rapidly  as 
possible,  and  the  chainmen  come  after,  lining  in  their  stakes 
by  the  eye  from  point  to  point.  The  whole  force  is  thus  kept 
pretty  steadily  in  motion. 

On  wide  plains,  a  set  of  chain-pins  may  be  used,  and  survey- 
stakes  placed  five  hundred  or  a  thousand  feet  asunder.  Yery 
often  stakes  at  intervals  of  two  hundred  feet  are  sufficient,  the 
levels  being  taken  every  hundred.  Location  stakes  are  put  in 
every  hundred  feet. 

4.  THE  TRANSITMAN  will  be  expected  to  keep  his  instru- 
ment in  adjustment,  and  to  be  quick  and  accurate  in  its  manip- 
ulation. It  is  not  needful  to  plant  it  as  if  for  eternity.  On 
the  contrary,  it  should  be  set  gently,  the  legs  thrust  but  slightly 
into  the  ground,  and  the  screws  worked  without  straining. 

On  long  tangents  it  is  a  good  plan  to  reverse  the  instrument 
at  each  new  point,  putting  the  north  and  south  ends  forward 
alternately.  Small  errors  in  adjustment  are  thus  balanced  in 
some  measure.  Select  also,  in  such  a  case,  some  distant  object 
in  range,  when  practicable,  to  run  by.  The  telescope,  in  wind 
or  sun,  will  sometim.es  warp  a  little  out  of  line. 

Never  omit  to  note  both  the  calculated  and  magnetic  bear- 
ings of  the  lines  on  survey,  and  of  -the  tangents  on  location. 
Guard  against  the  error  of  reading  deflections  or  bearings  from 
the  wrong  ten  mark ;  as,  for  instance,  34  instead  of  20. 

At  the  beginning  of  a  curve,  let  the  rear  chainman  know  the 
phis  of  the  P,  T,  Tell  the  front  chainman  the  degree  of  curve, 
and  instruct  him  how,  by  multiplying  1.75  by  the  degree,  he 
can  find  the  distance  of  each  full  station  from  the  range  of  the 
last  two.  A  quick  fellow  will  soon  pick  this  up,  and  become 
wonderfully  skilful  in  practice.  Thus  accomplished,  he  is  a 
check  on  wrong  deflections, 

In  running  curves,  a  tangential  angle  of  fifteen  degrees  from 
one  point  should  seldom  be  exceeded:  twenty  degrees  is  to  be 
regardetj  as,  a  maximum. 


SUGGESTIONS   CONCERNING  FIELD- WORK.  83 

Carry  a  pocket-compass,  and  observe  with  it  the  magnetic 
bearings  of  streams  and  roads  crossed. 

Record  daily  each  day's  run;  fill  out  the  distance  column, 
transcribe  the  chain-book,  and,  on  location,  record  the  apex 
distances  also  in  the  column  of  remarks. 

On  survey,  do  not  erase  from  the  field-book  the  notes  of 
abandoned  lines.  Simply  cancel,  and  mark  them  "  aban- 
doned," in  such  manner  that  they  may  still  be  legible. 

When  required  by  the  senior  assistant,  the  transitman  will 
aid  in  the  making  of  maps. 

5.  THE  LEVELLED  must  be  familiar  with  the  adjustments 
of  his  instrument,  keep  it  in  order,  and  handle  it  rapidly. 

On  survey,  establish  and  mark  benches  at  half-mile  intervals; 
on  location,  four  to  the  mile  when  practicable. 

Note  the  surface  elevations,  the  depths,  and  the  flood  heights, 
of  all  considerable  streams  crossed.  Take  a  rod  in  the  beds  of 
small  streams. 

Six  hundred  feet  each  way  should  be  regarded  as  the  maxi- 
mum sweep  of  the  level. 

Carry  a  hand-level,  and  thus  save  the  time  required  to  peg 
across  narrow  hollows,  or  over  heights  which  can  be  turned 
with  the  instrument. 

The  leveller  should  record  his  work,  and  make  up  the  profile 
daily. 

6.  THE  RODMAN  will  give  his  intermediates  close  by  the  sta- 
tions, observing  the  number  of   each   one  as  a  check  on  the 
chainmen,  and  calling  it  out  to  the  leveller.     He  should  have 
r*n  eye  to  abrupt  irregularities  in  the  ground,  arid  give  plus 
elevations  when  necessary. 

lie  will  keep  note  of  bench-marks  and  turning-pegs,  describ- 
ing the  latter  occasionally  with  reference  to  the  nearest  stake, 
that  the  levels  may  be  taken  up  speedily  in  case  of  a  revision 
of  the  line. 

When  unaccompanied  by  an  axeman,  the  rodman  is  equipped 
v.'ith  belt  and  hatchet.  Sometimes  he  is  furnished  also  with  a 
steel  pin  for  turning  on.  The  pin  has  a  ring  through  the  head, 
by  which  it  may  be  hung  to  a  spring  hook  in  the  belt. 

The  rodman  will  assist  the  leveller  at  record  and  profiles, 
and  transcribe  the  slope-book  daily. 

If  stakes  of  survey  are  set  at  intervals  of  two  hundred  feet, 
give  rods  every  hundred  feet,  as  nearly  as  the  midway  points 
can  be  guessed. 


84  THE  CURVE-PROTRACTOR. 

7.  THE  SLOPEMAN  will  give  backsights,  and  take  the  cross 
slopes  for  one  hundred  feet  on  each  side  of  the  line  at  everj 
station. 

8.  THE  REAR  CHAINMAN  will  carry  a  book  in  which  to  note 
the  turning-points,  the, crossings  of  roads,  streams,  swamps 
woodland,  and,  when  convenient,  property  lines  also.     He  wil 
hand  it  in  daily  to  the  transitman  for  record.     As  each  succes- 
sive chain  is  stretched,  the  rear  chainman  calls  out  the  numbei 
of  the  stake  it  is  stretched  from:  this  assures  the  selection  o] 
the  right  number  for  the  stake  ahead. 

9.  ONE  AXEMAX  will  be  detailed  to  make  stakes,  another  tr 
mark  and  drive  them.     Additional  axemen  may  be  employee 
at  the  discretion  of  the  senior  assistant,  as  the  work  requires 
them.     Wanton  destruction  of  timber,  fences,  growing  crops, 
or  other  property,  should  not  be  allowed.     Axemen  must  be 
careful,  in  passing  through  the  country,  to  do  as  little  damage 
as  possible. 


XXV. 

THE  CURVE-PROTRACTOR,   AND   THE   PROJECTING 
OF  LOCATIONS. 

1.  The  curve-protractor  is  simply  an  eight-inch,  semi-circu- 
lar horn  protractor,  upon  which  a  series  of  twenty-three  curves, 
from  half  a  degree  up  to  eight  degrees,  is  finely  engraved,  witt 
radii  of  400  feet  to  an  inch.  After  some  years'  use  in  his  owr 
practice,  the  contrivance  was  transmitted  by  the  writer  to  tin 
well-known  firm  of  James  W.  Queen  &  Co.,  mathematical- 
instrument  makers,  New  York  and  Philadelphia,  by  whom  ii 
is  now  manufactured.  It  greatly  facilitates  the  projecting  oj 
lines  and  solution  of  field-problems  on  location.  It  enables 
the  engineer,  for  example,  by  a  short,  graphical  process  ant 
rapid  inspection,  to  find  the  curve  which  shall  close  an  angk 
between  tangents,  or  terminate  a  compound  curve,  and  pass  al 
the  same  time  through  some  fixed  intermediate  point,  withoul 
liability  to  the  errors,  and  free  from  the  loss  of  time,  in  velvet 
in  a  tedious  calculation.  Other  applications,  such  as  the  nice 
adjustment  of  line  among  buildings,  on  precipitous  steeps,  and 
the  like,  will  suggest  themselves  to  the  experienced  reader. 


THIS   CURVE -PROTRACTOR.  85 

2.  For  office  use,  the  writer  prefers   a  home-made    curve- 
protractor  of  isinglass,  prepared  as  follows :  Take  a  thin,  clear 
sheet,  say  six  by  ten  inches,  free   from  bubbles  and  cracks. 
Block  it  securely  on  the  drawing-table  with  thumb-tacks,  set- 
ting the  shanks  close  against  the  edge  of  the  sheet,  but  not 
piercing  it,  and  the  heads  lapping  its  edge.     From  a  centre, 
midway  of  one  of  the  long  sides  and  near  its  margin,  strike  the 
curves  from  12°  or  less,  varying  outwards  by  half-degrees  to 
C°;  thence  by  quarter-degrees  to  4°;  and  thence  by  ten-minute 
differences  to  2^°.     This  covers  one  side  of  the  sheet,  the  scale 
being  400  feet  to  an  inch.     Xow  release  the  sheet,  turn  it  over, 
and  on  its  other  face  strike   the   remaining  curves,  down  to 
ten  minutes,  from  centres  on  the  table,  in  the  reverse  direc- 
tion, so  that  they  shall  cross  the  first  series  at  a  large  angle. 
Space  them  about  three-eighths  of  an  inch  asunder  at  the  mid- 
dle.    Use   a  needle-point  centre  for  the  first  series,  to  avoid 
boring  a  large  hole  in  the  sheet.     Add  also,  on  that  face,  two 
radial  lines  drawn  towards  the  corners.     Score  the  fractional 
curves  very  lightly,  the  full  figure  curves  a  little  deeper,  but 
all  of   them    with   steadiness    and    delicate    stress.     Practise 
beforehand  on  a  separate  slip,  for  the  right  intensity  of  stroke. 
Engrave  the  numbers  with  a  stiff  steel  point  on  the  opposite 
side  of  the  sheet  to  that  upon  which  the  corresponding  curve 
is  traced.     Bring  the  work  out  by  rubbing  it  with  India  ink. 
If  preferred,  the  flat  curves  on  the  reverse  side  may  be  colored 
with  carmine.     Duplicate  protractors  will  be  found  useful  in 
projecting  compound  curves.     Clip   off  the   four  corners,  re- 
enforce  the  edges  with  a  narrow  ribbon  of  tracing-linen,  folded 
over  them  and  glued  fast,  and  the  article  is  complete.     It  is 
perfect  for  its   use;   durable,  flexible,  spotlessly  transparent, 
not  liable  to  warp  or  change  dimensions  withwchanges  in  the 
temperature  or  moisture  of  the  air,  and,  withal,  takes  and  pre- 
serves a  visible  line,  thin  as  the  gossamer. 

3.  To  experienced  locating  engineers,  the   curve-protractor 
needs  no  wordy  commendation.     Contrasted  with  the  incon- 
venient  appliances   of  the   old  method, — cardboard,  veneer, 
glass,  or  dividers, — its  advantages  will  be   manifest.     A  few 
hints  as  to  the  manner  of  using  it  may  be  in  place. 

4.  First  of  all,  let  the  experimental  line  approximate  to  the 
probable  line   of  location ;  and,  upon  that  base,  construct  a 
contour  map,  with   reference   to   which   special   observations 
should  be  made  in  the  field,  and  the  chaining  done  with  care. 


86  THE  CURVE -PROTRACTOR. 

Extreme  accuracy  in  the  contours  need  not  be  attempted. 
Note  the  courses  of  streams,  ravines,  and  ridges,  the  average 
slopes  at  frequent  intervals,  and,  on  irregular  ground,  make 
illustrative  sketches  to  aid  in  utilizing  the  other  notes.  Prac- 
tice gradually  teaches  how  to  observe  critical  points  intelli- 
gently, and  to  record  them  briefly.  In  valleys  or  plains,  where 
the  location  indicated  is  made  up  of 'long  tangents  and  easy 
curves,  little  detail  is  required;  but  on  bluffy,  tortuous  ground, 
with  unavoidable  divides  to  overcome,  and  long  reaches  of 
maximum  gradient  to  be  fitted,  the  method  by  contours  is  not 
only  the  simplest  and  clearest  way  of  compiling  necessary 
information,  but  is  an  aid  to  the  engineer  in  projecting  the 
right  line,  which  no  substitute  can  fully  replace. 

5.  The  writer  is  forced  by  the  strong  constraint  of  experi- 
ence to  differ  on  this  subject  with  Mr.  Trautwine.     The  dif- 
ference,   however,    is    a    permissible    one,    and    implies    no 
lack  of   grateful    respect   for   that  veteran    engineer,   whose 
books  are  our  handy-books,  and  to  whose  genius  we  are  all 
debtors. 

6.  Having  made  the  map,  with  ten-foot  contours,  suppose, 
for  example,  that  a  continuous  gradient  five  miles  long  is  to  be 
located.     Spread  the  dividers  to  500  feet  by  the  scale,  start  at 
the  foot  of  the  ascent,  and  step  up,  complying  with  the  general 
trend  of  the  ground,  to  the  summit.     This  needful  preliminary 
gives  about  the  distance  you  have  to  work  on,  which  cannot 
in  many  cases  be  derived  from  the  experimental  line  directly. 
The  profile  furnishes  the  height  to  be  overcome;  and  you  are 
thus  prepared  to  assume  a  summit  cut,  and  determine  the 
gradient.     Having    adopted  one,    say,    of    66  feet  per    mile, 
observe   that  this   rises   five  feet  in  400    feet.      Spread    the 
dividers,  then,  to  400  feet  by  scale,  and  stand  one  leg  on  or 
near  the  summit,  at  a  point  corresponding  to  a  five  or  ten  unit 
in  the  elevation  of  the  gradient.     That  is  to  say,  if  the  grade 
elevation  at  the  summit  be  362,  for  instance,  stand  the  leg  of 
the  dividers  a  little  beyond  or  a  little  short  of  the  summit,  at  a 
point  where  the  grade  elevation  is  365  or  360.     Thence,  exer- 
cising good  judgment  to  conform  in  a  general  way  to  what  the 
location  ought  to  be,  and  to  make  no  angular  indirections  which 
cannot  be  closed  with  the  maximum  curvature,  step  forward 
down   the   incline.     Name  each  step  mentally  as  it  is  made, 
355,  350,  345,  340,  &e.,  and  spot  at  the  same  time  with  a  pencil- 
point  the  contour  or  half  space,  directly  opposite,  correspond- 


THE  CURVE -PROTRACTOR.  87 

ing  to  it  in  elevation.  Connect  the  pencil-marks  with  a  faint 
dotted  line. 

7.  Were  the  ground  a  straight,  regular  hillside,  the  steps 
would  be  made  directly  from  contour  to  half-space,  thence  to 
the  next  contour  below,  and  the  dotted  line  would  mark  out  a 
tangent  conforming  exactly  to  the  ground  surface.    On  devious 
slopes,  rounding  within  the  limit  of  the  sharpest  permissible 
curve,  the  same  exact  conformity  could  be  obtained,  if  desired, 
and   a  grade-line   laid  down  which  should  require  the  least 
possible  expense  in  building.     On  irregular,  winding  ground, 
an  approximation  only  to  the  dotted  line  can  be  made :  it  is 
nevertheless  a  guide  to  go  by;  and,  the  more  nearly  the  loca- 
tion project  approaches  it,  the  lighter  w,ill  the  work  of  con- 
struction be.     The  dotted  line,  in  short,  is  analogous  to  a 
profile ;  and  the  engineer  can  prescribe  his  cuts  and  fills  with 
reference  to  it,  by  means  of  curve  or  tangent,  just  as  on  the 
profile  he  does  the  same  by  means  of  grade-lines.     A  fairly 
correct  map  wrill  enable  him  to  construct  a  profile  from  the 
project,  and  to  amend  its  errors  without  the  trouble  and  ex- 
pense of  tentative  field-work.     The  writer's  habitual  practice 
has  been  to  base  his  preliminary  estimates  on  a  profile  thus 
deduced   from  the  map;   arid  he  recommends  the  practice  to 
others.     They  will  be  surprised  to  observe  the  likeness  between 
such  a  profile,  tolerably  well  done,  and  that  of  the  subsequent 
location. 

8.  It  is  a  good  custom,  and  one  which  cannot  prudently  be 
neglected  where  long  reaches  of  maximum  gradient  are  en- 
countered, to  "  slacken  grade  "  on  the  curves.     In  making  this 

.adjustment,  the  contour  map  is  exceedingly  useful.  An  ap- 
proximate project  is  first  required,  in  order  to  determine  the 
curvature,  and,  from  that,  the  varying  gradient.  The  location 
can  then  be  laid  down  on  the  map  with  satisfactory  precision. 
Opinions  differ  as  to  the  right  allowance  per  degree  of  curva- 
ture, and  no  experiments  seem  to  have  been  made  from  which 
to  deduce  an  authoritative  rule.  Some  say  0.025  per  degree 
per  100  feet;  others,  0.05;  others,  variously  between  the  two. 
Probably  0.05  is  the  safer  rate.  This  amounts  to  2.64  feet  on  a 
mile  of  continuous  one-degree  curve,  and  makes  a  nine-degree 
curve,  about  the  curve  of  double  resistance  at  ordinary  passen- 
ger speeds. 

9.  In  projecting  locations,  tiie  better  way  generally  ic  to 
strike  the  curves  first. 


88  THE   CURVE -PROTRACTOR. 

10.  The  following  tables  may  be  of  assistance.  It  was  need- 
ful, calculating  them  at  all,  to  calculate  them  right;  but  of 
course  such  exactness  as  the  figures  would  seem  to  indicate 
is  unattainable  in  practice. 


11.  TABLE  SHOWING  THE  DISTANCE,  D,  IN  FEET,  AT  WHICH 
A  STRAIGHT  LINE  MUST  PASS  FROM  THE  NEAREST 
POINT  OF  ANY  CURVE,  STRUCK  WITH  RADIUS  r,  IN 
ORDER  THAT  A  TERMINAL  BRANCH  HAVING  RADIUS 
K  =  2  r,  AND  CONSUMING  A  GIVEN  ANGLE,  X,  MAY 
MERGE  IN  SAID  STRAIGHT  LINE. 

D  =  (R  — r)  X  (1  — cos.  x). 


DEGREE  OF  THE  MAIN  CURVE. 

ANGLE 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

X. 

I). 

2° 

1.72 

1.15 

0.86 

0.69 

0.57 

0.49 

0.43 

0.38 

0.34 

3° 

4.01 

2.67 

2.00 

1.60 

1.34 

1.15 

1.00 

0.89 

0.80 

4° 

6.88 

4.58 

3.44 

2.75 

2.29 

1.96 

1.72 

1.53 

1.37 

5° 

10.89 

7.29 

5.44 

4.35 

3.63 

3.11 

2.72 

2.42 

2.18 

6° 

15.76 

10.50 

7.88 

6.30 

5.25 

4.50 

3.94 

3.50 

3.15 

7° 

21.49 

14.32 

10.74 

8.59 

7.16 

6.14 

5.37 

4.77 

4.30 

8° 

28.36 

18.91 

14.18 

11.35 

9.45 

8.10 

7.09 

6.30 

5.67 

9° 

35.24 

23.49 

17.62 

14.09 

11.75 

10.07 

8.81 

7.83 

7.05 

10° 

43.55 

29.13 

21.77 

17.42 

14.52 

12.44 

10.89 

9.68 

8.71 

THE  CURVE -PROTRACTOR.  89 

If  R  =  \\  r,  use  half  the  tabular  distance;  if  R  =  3  r,  use 
twice  the  tabular  distance;  if  R  =  4  r,  use  three  times  the 
tabular  distance,  and  so  on. 


12.  TABLE  SHOWING  THE  DISTANCE,  d,  IN  FEET,  AT  WHICH 
CURVES  OF  CONTRARY  FLEXURE  MUST  BE  PLACED 
ASUNDER  IN  ORDER  THAT  THE  CONNECTING  TANGENT, 
T,  MAY  BE  300  FEET.  LONG. 


DEGREE  OF  CURVE. 

DEGREE  OF  CURVE. 

DEGREE  OF  CURVE. 

1° 

2° 

3° 

4° 

5° 

6° 

7°          8° 

9° 

10° 

</. 

1° 
2° 
3° 
4° 
5° 
6° 

3.9 

5.24 

7.84 

5.92 
9.43 
11.77 

6.29 
10.38 
13.43 
15.65 

6.35 
11.20 
14.64 
17.39 
19.54 

6.68 
11.70 
15.68 
18.76 
21.22 
23.32 

6.86 
12.20 
16.45 
19.90 
22.76 
25.20 
27.25 

7.00 
12.55 
17.09 
20.82 
24.01 
26.70 
29.01 
31.05 

7.08 
12.80 
17.61 
21.64 
25.07 
28.00 
30.58 
32.82 
34.82 

7.18 
13.06 
18.05 
22.31 
25.97 
29.13 
31.93 
34.41 
36.31 
38.56 

1° 
2° 
3° 
4° 
5° 
6° 
7° 
8° 
9° 
10. 

8° 
9° 
10° 

•• 

•• 

Examples. 

A  7°  and  4°  should  be  19.9  feet  asunder;  a  5°  and  9°  should 
be  25.07  feet  asunder. 

As  approximations,  for  a  connecting  tangent  400  feet  long, 
take  twice  the  tabular  distance:  fora  connecting  tangent  200 
feet  long,  take  half  the  tabular  distance. 


PROBLEMS  IN  FIELD  LOCATION. 


XXVI. 

HOW  TO  PKOCEED  WHEN  THE    P.   C.   IS    INACCES- 
SIBLE. 

1.  Suppose,  for  example,  a  pro- 
jected 5°  curve,  beginning  at  stake 
24.20,  or  B  in  the  diagram. 

FIRST  METHOD.  —  At  any  point 
A,  which  we  will  assume  to  be 
stake  23.40,  set  up  the  transit.  Let 
it  be  judged  that  stake  27,  marked 
D  in  the  diagram,  must  fall  on  ac- 
cessible ground.  Then  the  distance 
B  D,  around  the  curve,  is  280  feet, 
corresponding  to  an  angle  E  B  D  of 
7°  at  the  circumference,  or  an  angle 
of  14°  at  the  centre.  The  chord  of 
a  1°  curve  consuming  this  angle,  by 
Table  XVI.,  is  1,396.6  feet;  that  of 

a  5°  curve,  B  D  in  the  figure,  is  one-fifth  of  this,  or  279.3  feet. 
In  the  triangle  A  B  D  are  thus  known  the  sides  A  B,  B  D,  and 
the  sum  of  the  angles  at  A  and  D,  which  sum  is  equal  to  the 
angle  E  B  D. 

Hence,  by  trigonometry,  — 


As  the  sum  of  the  sides  given  =  359.3  AC    . 
Is  to  their  difference  =199.3.     .. 

So  is  tan.     sum  of  anles  at  base   =  3°  30'  . 


To  tan.  |  their  difference 


=  1°  56^ 


7.444543 

2.299507 
8.786486 

8.530536 


Adding  half  the  difference  to  half  the  sum,  the  larger  angle, 
A,  is  found  to  be  5°  26£'  ;  subtracting  half  the  difference  from 
half  the  sum,  the  smaller  angle,  D,  is  found  fo  be  1°  33i'.  The 

93 


PROBLEMS  IN  FIELD  LOCATION. 


XXVI. 

HOW  TO  PROCEED  WHEN  THE    P.   C.   IS    INACCES- 
SIBLE. 

1.  Suppose,  for  example,  a  pro- 
jected 5°  curve,  beginning  at  stake 
24.20,  or  B  in  the  diagram. 

FIRST  METHOD.  —  At  any  point 
A,  which  we  will  assume  to  be 
stake  23.40,  set  up  the  transit.  Let 
it  be  judged  that  stake  27,  marked 
D  in  the  diagram,  must  fall  on  ac- 
cessible ground.  Then  the  distance 
B  D,  around  the  curve,  is  280  feet, 
corresponding  to  an  angle  E  B  D  of 
7°  at  the  circumference,  or  an  angle 
of  14°  at  the  centre.  The  chord  of 
a  1°  curve  consuming  this  angle,  by 
Table  XVI.,  is  1,396.6  feet;  that  of 

a  5°  curve,  B  D  in  the  figure,  is  one-fifth  of  this,  or  279.3  feet. 
In  the  triangle  A  B  D  are  thus  known  the  sides  A  B,  B  D,  and 
the  sum  of  the  angles  at  A  and  D,  which  sum  is  equal  to  the 
angle  E  B  D. 

Hence,  by  trigonometry,  — 


As  the  sum  of  the  sides  given  =  359.3  A  C    . 
Is  to  their  difference  =199.3.     .     . 

So  is  tan.  £  sum  of  angles  at  base   =  3°  30'  . 

To  tan.  i  their  difference  =  1°  56^ 


7.444543 

2.299507 
8.786486 

8.530536 


Adding  haK  the  difference  to  half  the  sum,  the  larger  angle, 
A,  is  found  to  be  5°  26^ ;  subtracting  half  the  difference  from 
half  the  sum,  the  smaller  angle,  D,  is  found  fo  be  1°  33^'.  The 

93 


94       HOW  TO  PROCEED  WHEN  THE  P.  C.  IS  INACCESSIBLE. 


length  of  the  side  A  D  may  be  found  in  like  manner  by  trigo- 
nometrical proportion;  or,  perhaps  more  simply,  thus:  — 

B  D  X  nat.  cos.  D  =  D  F  =  279.2. 
B  A  X  nat.  cos.  A  =  A  F  =  79.6. 
AF  +  FD  =  AD  =  358.8. 

We  are  now  prepared,  from  our  point  A,  to  deflect  the  angle 
5°  26$'  R,  and  lay  out  the  line  A  D  to  the  point  D  on  the  curve. 
Moving  the  instrument  to  that  point,  and  backsighting  to  A,  a 
deflection  of  1°  334-'  R  places  the  telescope  on  line  DB;  a  fur- 
ther deflection  of  7°  places  it  in  tangent  at  D,  and  the  curve 
may  thence  be  traced  in  both  directions. 

2.  SECOND    METHOD.  —  Having,    as    in    the    first  method, 
judged  that  stake  27,  marked  D,  must  fall  on  accessible  ground, 
and  thus  determined  the  central  angle  subtended  by  the  arc 
B  D,  refer  to  Table  XVI.  for  the  tangent  of  a  1°  curve,  corre- 
sponding to  14°,  the  given  angle.     It  proves  to  be  703.5  feet. 
One-fifth  of  this,  140.7  feet,  is  the  tangent  or  apex  distance, 
BC,  of  a  5°  curve,  which  may  be  measured  on  the  ground. 
Moving  the  instrument  to  C,  turning  14°  R,  and  laying  off  the 
line  C  D  =  B  C,  the  point  D  on  the  curve  is  ascertained. 

3.  The  preceding  methods  are  manifestly  applicable  to  the 
ends  also  of  curves,  as  well  as  the  beginnings.     A  case  not 

unfrequent  in  practice  may  be  added  in 
conclusion  of  the  subject. 

Suppose  a  2°  curve  terminating  at  C,  in 
marsh  or  stream  not  measurable  directly. 
Let  C  fall  at  stake  32.20.  At  any  con- 
venient point  A,  say  stake  29,  place  the 
transit  with  telescope  in  tangent.  The 
arc  A  C,  =  320  feet,  includes  an  angle  of 
6°  24'.  The  tangent  of  a  1°  curve  corre- 
sponding to  this  angle  in  Table  XYI.  is 
320.34  feet;  that  of  a  2°  curve  is  therefore 
160.2  =  A  B.  Move  to  B,  deflect  6°  24'  R 
into  the  range  of  the  terminal  tangent, 
and  fix  E  on  the  opposite  shore.  Fix  also 
D,  and  note  the  angle  E  B  D.  Move  to 
E.  Measure  the  angle  DEB,  and  the  distance  D  E.  The  tri- 
angle BED -may  then  be  solved.  If  B  E  is  found  to  be  670 
feet,  C  E  =  670  —  160.2  =  509.8,  and  stake  E  =  32.20  +  509.8, 
=  say  37.30. 


IfOW  TO  PROCEED  WHEN  THE  P.  C.  C.  18  INACCESSIBLE.     95 


XXVII. 

HOW   TO  PROCEED  WHEN  THE  P.  C.  C.    IS   INAC- 
CESSIBLE. 

1.  Suppose    a    4°   curve, 
A  B,  compounding  at  B  into 
a  G°  curve  B  C. 

FIKST  METHOD.  —  Place 
the  transit  at  any  point  A, 
say  stake  34.  Let  the  pro- 
posed P.  C.  C.  fall  at  stake 
36.25.  Assume  that  we  wish 
to  reach  C,  on  the  second 
curve,  by  means  of  the 
straight  line  ADC.  The 
arc  A  B,  covering  225  feet 
of  a  4°  curve,  subtends  an 
angle  of  9°.  A  D  is  half  the  chord  of  twice  this  angle. 

By  Table  XVI.,  the  chord  of  18°  on  a  1°  curve  is  1,792.7  feet. 
That  of  a  4°  curve  is  therefore  448.2  feet,  half  of  which  = 
224.1,  =  A  D.  The  versin.  of  18°  on  a  1°  curve,  by  the  same 
table,  is  70.54  feet;  one-fourth  of  which,  or  17.635,  is  the  versin. 
B  D,  corresponding  to  the  same  angle  on  a  4°  curve.  In  order 
to  find  what  angle  on  the  6°  curve  this  versin.  B  D,  =  17.635 
feet,  corresponds  to,  multiply  it  by  6,  and  seek  the  product, 
105.81,  in  Table  XVI.,  where  it  is  found,  nearly  enough  for 
field-practice,  opposite  the  angle  22°  04'.  The  chord  of  that 
angle,  on  a  1°  curve,  is  seen  at  the  same  time  in  the  adjoining 
column  to  be  2,193.2  feet;  on  a  6°  curve  it  is  therefore  365.5 
feet,  one-half  of  which,  =  182.75  feet,  =  D  C,  and  one-half  of 
22°  04'  =  11°  02',  =  the  angle  covered  by  the  arc  B  C.  Thus 
are  found  the  angle  at  A  =  9°,  the  angle  at  C  =  11°  02',  and 
the  distance  AC  =  224.1  -f  182.75,  =  406.85  feet.  The  angle 
11°  02' corresponds  to  a  length  of  1.84  feet  on  the  6°  curve; 
C,  therefore,  falls  at  stake  36.25  +  1.84  =  38.09.  With  these 
data  the  field-work  is  obvious. 

2.  SECOND  METHOD.  —  Having  reached   the  point   A,  and 
determined  the  arc  A  B  =  9°,  as  above,  find  in  Table  XVI.  the 
tangent  450.95  feet,  corresponding  to  the  given  arc,  one-fourth 


96 


TO  SHIFT  A  P.  C. 


of  which  =  112.7  feet,  =  tan.  AE  for  the  4°  curve.  Move  to 
E,  deflect  9°  R ;  range  out  the  line  E  F,  made  up  of  E  B  =  A  E 
=  112.7  feet,  and  BF  any  convenient  distance,  say  90  feet. 
This  90  feet  is  the  assumed  tangent  of  some  unknown  angle  on 
the  6°  curve.  To  find  the  angle,  multiply  90  hy  G,  and  seek  the 
product,  540,  in  the  tan.  column  of  Table  XVI.,  where  it  is 
found  opposite  10°  46'.  By  moving  then  to  F,  deflecting  10° 
46'  R,  and  measuring  F  C  =  90  feet,  the  point  C  is  fixed  on  the 
second  curve. 

3.  Should  unexpected  obstacles  be  met  in  carrying  out  either 
of  these  plans,  the  triangles  AGC  or  EGF  may  be  solved, 
and  the  point  C  fixed  by  means  of  the  lines  A  G,  G  C. 

4.  The   application   of    the  foregoing  methods    to    Burning 
obstacles  on  simple  curves  needs  no  special  instance. 


XXVIII. 

TO    SHIFT  -A    P.  C.    SO   THAT  THE    CURVE    SHALL 
TERMINATE  IN  A   GIVEN  TANGENT. 

1.  Suppose  a  3°  curve  AB  to 
have  been  located,  containing  an 
angle  of  44°  2&,  and  ending  in 
tangent  B  E :  required,  that  it 
shall  end  in  tangent  D  F,  parallel 
to  B  E.  It  is  plain,  from  the 
diagram,  that  if  the  curve  and 
its  initial  tangent  be  moved  forward',  like  the  blade  of  a  skate, 
until  the  germinal  tangent  merges  in  D  F,  the  P.  T.  will  have 
traversed  the  line  B  D,  equal  and  parallel  to  A  C.  If,  there- 
fore, on  the  ground  at  B,  the  angle  E  B  D,  equal  to  the  whole 
angle  consumed  by  the  curve,  in  this  case  44°  26',  be  laid  off 
to  the  right,  and  the  distance  B  D  to  the  range  of  the  proposed 
terminal  tangent  be  measured,  the  equal  distance  AC,  from 
the  original  to  the  required  P.  C.,  is  thus  directly  ascertained. 

Should  such  direct  measurement  be  impracticable,  range  out 
the  tangent  BE,  and,  at  any  convenient  point,  measure  the 
distance  from  it  square  across  to  the  proposed  terminal  tan- 
gent D  F,  say  56  feet.  Then  in  the  right  triangle  BED,  mak- 
ing BD  radius,  we  have  given  the  angle  at  B  =  44°  26',  and 


TO  SUBSTITUTE  A    CURVE  OF  DIFFERENT  RADIUS.       97 

the  sine  E  D  =  50  feet.  Hence,  by  trigonometry,  E  D  -f-  nat. 
sin.  44°  2(>',  or  56  -f-  0.7,  =  B  D  =  80  feet,  =  distance  A  C  along 
the 'initial  tangent,  from  the  erroneous  to  the  correct  P.  C. 

2.  This  problem  occurs  more  frequently  than  any  other  in 
the  field;  and  the  young  engineer  should  have  it  by  heart,  that 

•the  distance  square  across  between  terminal  tangents,  divided 
by  the  natural  sine  of  the  total  angle  turned,  will  give  him  the 
distance  he  is  to  advance  or  recede  with  his  P.  C.  to  make  a  fit. 

3.  Excepting  on  precarious  rocky  steeps,  city  streets,  or  like 
exact  confines,  to  strike  within  two  feet  of  any  point  desig- 
nated in  the  project,  may  be  considered  striking  the  mark. 
Astronomical  nicety,  whether  v/ith  transit  or  level,  in  an  ordi- 
nary railroad  location,  is  mere  waste  of  time. 

4.  The  observant  reader  will  not  fail  to  perceive  that  the 
foregoing  rule  applies  to  systems  of  curves,  or  to  compound 
lines  also,  the  angle  E  B  D  being  the  angle  included  between 
the  initial  and  terminal  tangents,  let  what  flexures  or  indirec- 
tions soever  have  been  interposed ;  and  that,  if  the  angle  re- 
ferred to  be  either  180°  or  360°,  adjustment  by  shift  of  P.  C.  is 
impracticable.     In  those  cases,  a  change  of  radius  becomes 
necessary. 


XXIX. 

TO  SUBSTITUTE  FOR  A  CURVE  ALREADY  LOCATED, 
ONE  OF  DIFFERENT  RADIUS,  BEGINNING  AT  THE 
SAME  POINT,  CONTAINING  THE  SAME  ANGLE,  AND 
ENDING  IN  A  FIXED  TERMINAL  TANGENT. 

1.  Suppose  the  4°  curve  AB, 
containing  an  angle  of  32°  20',  to 
have  been  located,  and  that  it  is 
required  to  substitute  for  it  an- 
other curve  AC,  which  shall  end 
in  a  parallel  tangent  C  F,  60  feet 
to  the  right. 

FIRST  M  E  T  H  o  D.  —  Find  the 
length  of  the  long  chord  A  C,  = 

A  B  -{-  B  C.     Referring  to  Table  XVI.,  the  chord  of  a  1°  curve 
for  32°  20'  is  seen  to  be  3,190.8  feet;  that  of  a  4°  curve,  there- 


98  TO  FIND   THE  POINT  AT   WHICH  TO   COMPOUND. 

fore,  =  797.7  feet,  say  798  feet,  =  A  B.  To  find  B  C,  solve 
the  triangle  BDC,  observing  that  the  angle  DBC  =  B  AI  = 
one-half  of  the  central  angle  32°  20',  =  16°  10',  and  that  t>  C 
=  00  feet.  Then  D  C  -f-  nat.  *in.  16°  10'  =  60  -=-  .278  =  say 
216  feet,  =  B  C.  Hence  AC  =  AB-fBC  =  798  -f  216  = 
1,014  feet. 

Having  thus  found  the  length  of  chord  A  C,  the  radius  and 
rate  of  curvature  may  be  deduced  as  in  X. 

Or,  dividing  the  tabular  chord  of  32°  20'  by  chord  A  C  = 
1,014,  the  degree  of  the  required  curve  is  ascertained  directly 
to  be  3.15,  equivalent  to  3°  09'. 

2.  SECOND  METHOD.  —  Find  the  apex  distance  AH,  =  AI 
+  I  H.  The  tabular  tangent  of  32°  20'  divided  by  4  gives  A  I 
=  415  feet.  In  the  triangle  KDC,  the  side  DC  -i-  nat.  sin. 
K  =  60  -f-  nat.  sin.  32°  20V  =  112  feet  =  KG  =  I  H.  Then 
AH  =  A I  -f-  IH  =  415  +  112  =  527  feet;  and  the  tabular 
tangent  1,661  -f-  527  gives  3.15,  equivalent  to  3°  09',  the  degree 
of  the  required  curve  A  C,  as  before. 


XXX. 

HAVING  LOCATED  A  CURVE  A  B  C,  TO  FIND  THE 
POINT  B  AT  WHICH  TO  COMPOUND  INTO  ANOTHER 
CURVE  OF  GIVEN  RADIUS,  WHICH  SHALL  END  IN 
TANGENT  E  F,  PARALLEL  TO  THE  TERMINAL 
TANGENT  OF  THE  ORIGINAL  CURVE,  AND  A  GIVEN 
DISTANCE  FROM  IT. 

1.  To  find  B,  the  angle  BIG  must  be 
found.  Call  the  given  distance  between 
tangents  D;  the  larger  radius,  K;  the 
smaller  one,  r;  the  required  angle,  a. 
Then,  referring  to  the  figure,  observe 
that  in  the  triangle  I H  K,  I  H  being  ra- 
dius, I  K  is  the  cosine  a;  i.e.,  I  K  -f-  1 II 
=  nat.  cosine  a.  But  I II  =  R  —  r;  I  K 
=  I  C  —  K  C,  and  K  C  =  K  F  or  II  E 

-f  F  C,  =  r  -f-  D;  i.e.,  1  K  =  R  —  r  —  D.     Hence  'nat.  cosine 
a  =  R  —  r  —  D,  -i-  K  —  r  =  1  —  11)  -f-  (H  —  r)]. 


TO   SHIFT  A   P.  C.  C.  99 

The  same  reasoning  would  apply  if  A  BE  were  the  curve 
first  located,  and  a  terminal  curve  of  larger  radius  required  to 
be  put  in. 

2.  We  have,  then,  the  following  general  rule  for  such  cases: 
Divide  the  perpendicular  distance  between  terminal  tangents 
by  the  difference  of  the  radii,  and  subtract  the  quotient  from 
unity;  the  remainder  is  the  natural  cosine  of  the  angle  of  re- 
treat along  the  located  curve  to  the  required  P.  C.  C. 

Example. 

3.  A  3°  curve  on  the  ground,  to  find  the  P.  C.  C.  of  a  5°  curve 
striking  27  feet  to  the  right.     Here  D  =  27;  R  —  »•  =  1,910  — 
1.146,  =  764;  D  ~  R  —  r  =  27  -7-  784,  =  .03534;  and  1  — 
.03534  =  .96466  =  nat.  cosine  15°  17'.     We  must  go  back, 
therefore,  509  feet  on  the  3°  curve,  to  compound  into  the  5° 
curve.     Had  the  5°  curve  been   located  first,  we  must  have 
gone  back  306  feet  to  begin  the  3°  curve  which  should  strike  27 
feet  to  the  left.     In  either  case,  time  might  be  saved  by  moving 
directly  from  E  to  C,  or  the  reverse,  and  spotting  in  the  curve 
backwards.     To  do  this,  we  have  in  the  right  triangle  F  E  C, 
the  angle  E  =  half  of  15°  19',  =  7°  38|',  and  the  side^FC  =  27 
feet.     Then  E  C  =  27  -f-  nat.  sin.  7°  38*',  =  203  feet ;  and  if  E 
were  stake  54.20  on  the  5°  curve,  B  would  fall  at  stake  54.20 
_  3.06,  =  51.14;  and  C,  the  P.  T.  of  the  3°  curve,  at  51.14  + 
5.09,  =  stake  56.23. 


XXXI. 

TO  SHIFT  A  P.  C.  C.,  SO  THAT  THE  TERMINAL 
BRANCH  OF  THE  CURVE  SHALL  END  IN  A 
GIVEN  TANGENT. 

FIRST  CASE:  the  terminal  branch 
having  the  shorter  radius. 

1.  Suppose  the  compound  curve 
ACN  located,  and  that  it  is  required 
to  fix  a  new  P.  C.  C.  at  B,  from  which 
the  terminal  branch  BM  shall  merge 
in  tangent  M  L,  a  given  distance  from 
NO.  To  fix  B,  the  central  angle 
B  H  M  of  the  new  terminal  branch 
must  be  found,  and  substituted  for 
CIN.  Call  the  longer  radius  R;  the  shorter  one,  r;  the  dis- 


100  TO   SHIFT  A   P.  C.  C. 

tance  asunder  of  the  terminal  tangents,  D ;  the  central  angle, 
CIN,  =  IEK,  of  the  located  terminal  branch,  b;  and  the 
central  angle,  B  H  M,  =  HE  F,  to  be  substituted  for  it,  «. 

In  the  right  triangle,  EIK,  EK  =  EI  cos.  I E  K  =  (R  —  r) 
cos.  b. 

In  the  right  triangle  H  F  E,  E  F  =  E  II  cos.  II E  F  =  (II  —  r) 
cos.  a. 

Also,  F  K  =  L  O  =  D,  since  each  is  equal  to  r  —  K  L. 

ThenE  F  =  E  K  — FK;  i.e.,  (R  — r)  cos.  a=  (K  — r)  cos.  b 
—  D. 

Hence  nat.  cosine  a  =  nat.  cosine  b —  [D-f-  (R —  r)j. 

Were  the  curve  B  M  located,  and  the  curve  C  N"  to  be  substi- 
tuted for  it,  —  that  is  to  say,  were  a  given  and  b  required, — 
we  should  have,  by  transposition,  nat.  cos.  b  =  nat.  cos.  a  -f- 
[D-MR-r)] 

Example. 

A  3°,  compounding  into  a  5°  curve  at  C,  which  consumes  an 
angle  CIN,  =  30°  22',  and  ends  in  a  tangent,  NO,  which  is 
found,  by  measurement  of  L  O,  to  be  34  feet  too  far  to  the  left. 

Here,  *D  =  34,  R  =  1,910,  r  ==  1,146,  b  =  30°  22';  and,  by 
the  solution,  nat.  cos.  a  ==  nat.  cos.  30°  22'  —  (34  -f-  [1,910  — 
1,146])  =  0.8628  —  (34  -4-  704). 

34 log.  1.531479 

764     ,  lo^;.  2.883093 


.0445 log.  2.648386 

Then  0.8628  —  0.0445  =  0.8.183  =  cos.  35°  05',  =  angle  a  ; 
a— 6  =  BUM  —  C IN = EEC 
=  the  angle  of  retreat  from  the 
erroneous  P.  C.  C.  =  35°  05'  - 
30°  22'  =  4°  43',  equivalent  to  LVT 
feet,  on  the  3°  curve,  from  C'to  B. 
2.  SECOND  CASE:  the  terminal 
branch  having  the  longer  radius. 

Let  B  N  represent  the  terminal 
branch  located  with  central  angle 
I K  O  =  b,  and  suppose  it  required 
to  determine  the  new  arc  CM, 

with  central  angle  I  E  F  =  a.     Call  the  longer  radius  R,  the 
shorter  one  r;  the  distance  L  N  between  tangents,  D.     In  the 


TO  SHIFT  A  P.  C.  .C.  1  yik 

right  triangle  IKO,  KO  =  KI,  cofc  IK/0)  ±=  (R>-  V);  233.  <>. 
In  the  right  triangle  FIE,  EF  =  ET,  cos.  1F.F  ='  (R  —  r) 
cos.  a.  Also,  E II  =  L  N  =  D,  since  each  is  equal  to  R  —  K  L. 
Then  E  F  =  E  II  +  H  F  =  E  II -f  K  O;  i.e.,  (R  —  r)  cox. 
«=(R — r)  cos.  b  +  D.  Hence  nat.  cos.  a  =  nat.  cos.  b  -f 

iD-j-(R-r)]. 

Were  the  curve  C  M  located,  and  the  curve  B  N  to  be  sub- 
stituted for  it,  that  is  to  say,  were  a  given  and  6  required, 
we  should  have,  by  transposition,  nat.  cos.  b  =  nat.  cos.  a  — 
[D-MR-r)]. 

Example. 

A  5°  compounding  into  a  3°  curve  at  B,  which  consumes 
an  angle  of  44°  20',  and  terminates  at  N,  28  feet  too  far  to 
the  left.  Here  D  =  28,  R  =  1,910,  r  =  1,146,  b  =  44°  20; 
and,  by  the  solution,  nat.  cos.  a  =  nat.  cos.  44°  20'  -|-  (28 
-f-  764).  The  nat.  cos.  44°  20'_=  0.69883;  28  -f-  764  =  log. 
1.447158  —  log.  2.883093  =  log.  2.564065,  corresponding  to  the 
decimal  0.03665,  which,  being  added  to  nat.  cos.  44°  20;,  gives 
0.73548,  the  nat.  cos.  42°  29'.  Then  B  K  N  —  C  E  M  =  44°  20' 
—  42°  29'  =  1°  51'  =  angle  B I  C,  equivalent  on  a  5°  curve  to 
37  feet,  which  therefore  is  the  distance  around  the  arc  from 
B,  the  erroneous  P.C.C.,  to  C,  the  correct  one. 

3.  From  these  formulas  the  following  general  rule  may  be 
drawn:  Divide  the  distance  between  terminal  tangents  by  the 
difference  of  the  radii,  and  call  the  quotient  Q.  Find  the  nat- 
ural cosine  of  the  terminal  arc  already  located,  and  call  it  C. 
The  sum  or  the  difference  of  Q  and  C  will  be  the  natural 
cosine  of  the  terminal  arc  to  be  substituted  for  that  already 
located.  With  radii  in  the  order  R,  r,  should  the  terminal 

tangent  located  strike    ]  (    the  proposed  tangent;  or, 

'    OUtSlQG    ' 

with  radii  in  the  order  r,   R,    should  the  terminal  tangent 
located    strike  |        . ,      >    the   proposed  tangent,  —  take  the 

i  difference  I  of  Q  and  °  for  the  re^uired  cosine' 


100  TO  SHIFT  A   P.  C.  C. 

tance  asunder  of  the  terminal  tangents,  D ;  the  central  angle, 
C I N,  =  I E  K,  of  the  located  terminal  branch,  b  ;  and  the 
central  angle,  B  H  M,  =  HEF,  to  be  substituted  for  it,  «. 

In  the  right  triangle,  EIK,  EK  =  EI  cos.  I E  K  =  (R  —  r) 
cos.  b. 

In  the  right  triangle  H  F  E,  E  F  =  E II  cos.  II E  F  =  (II  —  r) 
cos.  a. 

Also,  F  K  =  L  O  =  D,  since  each  is  equal  to  r  —  K  L. 

ThenE  F  =  E  K  — FK;  i.e.,  (R  — r)  cos.  a=(R—-r)  cos.  b 
—  D. 

Hence  nat.  cosine  a  =  nat.  cosine  b  —  [D  -f-  (R  —  >•) ]. 

Were  the  curve  B  M  located,  and  the  curve  C  N"  to  be  substi- 
tuted for  it,  —  that  is  to  say,  were  a  given  and  b  required, — 
we  should  have,  by  transposition,  nat.  cos.  b  =  nat.  cos.  a  -f- 
ID-MR  — r)] 

Example. 

A  3°,  compounding  into  a  5°  curve  at  C,  which  consumes  an 
angle  CIN,  =  30°  22',  and  ends  in  a  tangent,  NO,  which  is 
found,  by  measurement  of  L  O,  to  be  34  feet  too  far  to  the  left. 

Here,  *D  =  34,  R  =  1,910,  r  =  1,146,  b  =  30°  22';  and,  by 
the  solution,  nat.  cos.  a  =  nat.  cos.  30°  22'—  (34  -f-  [1,910  — 
1,146])  =  0.8628  —  (34  -f-  704). 

34 log.  1.531479 

764     ,  IK.  2.883093 


.0445 log.  2.648386 

Then  0.8628  —  0.0445  =  0.8183  =  cos.  35°  05',  =  angle  a ; 
a  —  6  =  BIIM  —  CIN=  BEC 
=  the  angle  of  retreat  from  the 
erroneous  P.  C.  C.  =  35°  05'  - 
30°  22'  =  4°  43',  equivalent  to  157 
feet,  on  the  3°  curve,  from  C'to  B. 
2.  SECOND  CASE:  the  terminal 
branch  having  the  longer  radius. 

Let  BN  represent  the  terminal 
branch  located  with  central  angle 
I K  O  =  6,  and  suppose  it  required 
to  determine  the  new  arc  CM, 

with  central  angle  I  E  F  =  a.     Call  the  longer  radius  R,  the 
shorter  one  r;  the  distance  L  N  between  tangents,  D.     In  the 


TO   SHIFT  A   P.  C.  .;<7.  :   ;  v    '.    '.'  1(,jlv 

right  triangle  IKO,  KO  =  KI,  coS.  IK/C)  ±=  (B  —  r).  35.s.  «\ 
In  the  right  triangle  FIE,  EF  =  El,  cos.  IF, F  ='  (R  —  r) 
cos.  a.  Also,  E  H  =  L  N  =  D,  since  each  is  equal  to  R  —  K  L. 

Then  EF  =  EH  +  HF  =  EH  +  KO;  i.e.,  (R  — r)  cos. 
rt  =  '(R — r)  cos.  b  +  D.  Hence  nat.  cos.  a  =  nat.  cos.  b  -f 
[D--(R-r)]. 

Were  the  curve  C  M  located,  and  the  curve  B  N  to  be  sub- 
stituted for  it,  that  is  to  say,  were  a  given  and  b  required, 
we  should  have,  by  transposition,  nat.  cos.  b  =  nat.  cos.  a  — 
P-MR-r)]. 

Example. 

A  5°  compounding  into  a  3°  curve  at  B,  which  consumes 
an  angle  of  44°  20',  and  terminates  at  N,  28  feet  too  far  to 
the  left.  Here  D  =  28,  R  =  1,910,  r  =  1,146,  b  =  44°  20; 
and,  by  the  solution,  nat.  cos.  a  =  nat.  cos.  44°  20'  -(-  (28 
-f-  764).  The  nat.  cos.  44°  20'_=  0.69883;  28  -f-  764  =  log. 
1.447158  —  log.  2.S83093  =  log.  2.564065,  corresponding  to  the 
decimal  0.03665,  which,  being  added  to  nat.  cos.  44°  20',  gives 
0.73548,  the  nat.  cos.  42°  29'.  Then  B  K  N  —  C  E  M  =  44°  20' 
—  42°  29'  =  1°  51'  =  angle  BIG,  equivalent  on  a  5°  curve  to 
37  feet,  which  therefore  is  the  distance  around  the  arc  from 
B,  the  erroneous  P.C.C.,  to  C,  the  correct  one. 

3.  From  these  formulas  the  following  general  rule  may  be 
drawn:  Divide  the  distance  between  terminal  tangents  by  the 
difference  of  the  radii,  and  call  the  quotient  Q.  Find  the  nat- 
ural cosine  of  the  terminal  arc  already  located,  and  call  it  C. 
The  sum  or  the  difference  of  Q  and  C  will  be  the  natural 
cosine  of  the  terminal  arc  to  be  substituted  for  that  already 
located.  With  radii  in  the  order  R,  r,  should  the  terminal 

tangent  located  strike    j  £    the  proposed  tangent;  or, 

with  radii  in  the  order  r,  R,  should  the  terminal  tangent 
located  strike  <  . ,  >  the  proposed  tangent,  —  take  the 

i  difference  I  of  Q  and  C  for  the  re(luired  cosine- 


104 


TO  LOCATE  A    Y. 


son  that  each  is  equal  to  D  E.  Make  G  F  =  R  -4-  r,  the  diame- 
ter of  a  semicircle.  Said  semicircle  touches  tangent  13  A  at  D, 
its  middle  point;  and  D  E  being  perpendicular  to  G  F,  we  have 
by  geometry  GE:DE::DE:EF;  i.e.,  GE  X  EF,  or  R  X 
r,  =  D  E2.  Hence  D  E  =  B  D  =  U  A  =  y/R  X  r  =  R  tan.  i  x, 
and  we  are  thus  enabled  to  fix  the  points  E  and  A. 

3.  In  the  two  foregoing  problems,  the  angle  consumed  by 
curve  E  A  is  =  180°  —  x. 

Example. 

BE,  a  2^°  curve  located;  BA,  a  tangent:  to  complete  the 
Y  with  a  6°  curve,  E  A. 

By  the  first  method,  cos.  x  =  (R  —  »•)  -f-  (R  -f  r)  =  (2,292 
—  955)  H-  (2,292  -f  955)  =  1,337  -f-  3,247  =  log.  3.126131  - 
log.  3.511482  =  1.014649,  which  corresponds  to  log.  cos. 
9.614649,  or  to  the  decimal  number  0.4118,  indicating  in  either 
case  the  angle  65°  41'  =  x. 

DE  =  BD  =  DA  =  R  tan.  $x  =  2,292  X  0.6455  =  1,479.4. 
DE  may  be  found  also  by  reference  to  Table  XVI.,  where  the 
tangent  of  a  1°  curve  for  65°  41'  is  seen  to  be  3,698.6.  Dividing 
this  number  by  2|,  we  have  1,479.4,  as  above. 

Or,  by  the  second  method,  — 


D  E 


X  r 


=  \/2  188860  =  1  ,479.4. 


Having  thus  the  means 
of  fixing  points  E,  D,  and 
A,  the  curve  E  A  can  be 
laid  down. 

4.  If  B  A  is  curved  con- 
vex to  the  Y,  construct 
the  figure  as  in  margin, 
and  reason  thus:  — 

In  the  triangle  EGF, 
formed  by  lines  connect- 
ing the  curve-centres,  the 
sides  are  respectively 
equal  to  the  sums  of  the 
contiguous  radii:  the 
angles  may  therefore  be 
found  as  in  Case  III., 
Trigonometry. 
Lines  drawn  bisecting  the  central  angles  of  the  severaJ 


TO  LOCATE  A    Y.  105 

curves  will  pass  through  the  points  of  intersection  of  the  tan- 
gents to  those  curves,  severally.  But  lines  so  drawn  in  this 
case  bisect  also  the  angles  of  a  triangle,  and,  demonstrably 
by  geometry,  meet  in  one  point  equidistant  from  the  three 
sides  of  the  triangle.  That  point,  therefore,  must  be  a  com- 
mon P.  I.  for  all  the  curves,  and  that  equidistance  the  "tan- 
gent "  length  common  to  them  all. 

Example. 

Given  B  A,  a  3°,  and  B  C,  a  4°  curve:  to  complete  the  Y  with 
a  5°  curve,  C  A. 

E  F  =  1,910  +  1,146  =  3,056. 
GF  =  1,433  +  1,146  =  2,579. 
E  G  =  1,910  -|-  1,433  =  3,343. 

Then,  by  Case  III.,  Trigonometry, — 

As  EG,  3,343.     .     .     .     log.  (a.  c.)  6.475864 

Is  to  E  F  +  G  F,  5,635  .'    .     .     .     log.     .     .     3.750894 
So  is  E  F  —  G  F,     477   .     .     .     .     log.     .     .    2.678518 

To  diff.  of  segments  of  E  G,  804 2.905276 

Adding  half  the  difference  to  half  the  sum  of  the  segments 
of  the  base  EG,  we  shall  have  the  greater  of  them;  i.e., 
(3,343  +  804)  -:-  2  =  2,073.5,  which  is  the  cos.  E,  E  F  being 
radius.  Hence  2,073.5  -=-  3,056  =  log.  3.316704  —  log. 
3.485153  =  9.831551  =  cos.  47°  16'  =  E.  By  Table  XVI.,  the 
tangent  of  a  1°  curve  corresponding  to  this  angle  is  2,507.3: 
that  of  a  3°  curve,  therefore,  is  835.8  =  the  common  tangent 
B  Dor  DA.  Multiplying  the  common  tangent  by  4,  we  shall 
find  opposite  the  product  in  Table  XVI.  the  central  angle  of 
the  4°  curve  to  be  60°  32';  multiplying  it  by  5,  we  find,  in  like 
manner,  the  central  angle  of  the  5°  curve  to  be  72°  12'.  Arc 
B  A,  =  47°  16',  is  equivalent  to  1,575  feet  on  the  3°  curve;  arc 
B  C,  =  60°  32',  is  equivalent  to  1,513  feet  on  the  4°  curve. 
Points  being  thus  fixed  at  A  and  C,  curve  C  A  can  be  laid  on 
the  ground. 

5.  //  curve  B  A  is  concave  to  the  Y,  the  radii  being  given, 
construct  the  figure  as  follows:  — 

First  draw  the  triangle  GFE,  the  sides  of  which  are  obvi- 
ously derived  from  the  given  radii.  Prolong  the  sides  E  G  and 
E  F  indefinitely.  Bisect  the  exterior  angles  at  G  and  F  with 


106 


TO  LOCATE  A    Y. 


lines  meeting  at  D,  and  from  D  let  fall  perpendiculars  on  EB, 
EA,  and  GF.  Then,  comparing  triangles  GBD,  GCD,  the 
angles  at  G  are  equal  by  construction ;  the  angles  at  13  and  C 
are  right  angles,  the  side  G  D  common.  Hence  the  triangles 
are  equal  in  all  their  parts:  B  G  =  G  C,  and  B  D  =  D  C.  By 
like  reasoning,  it  appears  that  CF  =  FA,  and  DA  =  DC. 
The  point  D  being  equidistant  from  the  right  lines  E  B,  EA, 
which  limit  angle  E,  a  line  bisecting  that  angle  will  strike 
point  D. 


6.  It  may  be  remarked,  therefore,  that  lines  bisecting  the 
vertical  angle  and  the  exterior  angles  contained  between  the 
base  and  the  prolongation  of  the  sides  of  any  triangle,  will 
meet  in  a  point  equidistant  from  the  base  and  the  said  prolon- 
gations. We  thus  have  in  the  figure  all  the  conditions  for  fit- 
ness of  the  curves.  It  remains  only  to  solve  the  triangle  G  F  E, 
seeing  that  from  its  angles  the  required  central  angles  can  be 
obtained. 

Example. 

B  A,  a  1°,  B  C,  a  6°  curve,  located :  to  complete  the  Y  with 
an  8°  curve,  C  A. 


TO   LOCATE  A    TANGENT  TO  A    CURVE.  107 

111  triangle  G  F  E,  — 

E  F  =  5,730  —  717  =  5,013. 
E  G  =  5,730  —  955  =  4,775. 
G  F  -  955  +  717  =  1,672. 

Then,  by  Case  III.,  Trigonometry,  — 

AsEF  .  .  .  5,013  ....  log.  (a.  c.)  6.299902 
Is  to  E  G  -f  G  F,  6,447  ....  log.  .  .  3.809358 
SoisEG  —  GF,  3,103  .  .  .  .  log.  .  .  3.491782 

To  cliff,  seg.  of  base,  3,991  .     .     .     log.     .     .     3.601042 

The  longer  segment,  therefore,  is  4,502;  the  shorter,  511. 
Cos.  E  =  the  longer  segment  divided  by  E  G  =  4,502  -4-  4,775  = 
log.  3.653405  —  3.678973  =  9.974432  =  cos.  19°  28'  =  angle  E. 

Cos.  G  F  E  =  the  shorter  segment  divided  by  GF  =  511  -j- 
1,672  =  log.  2.708421  —  log.  3.223236  =  9.485185  =  cos.  72° 
12'  =  angle  G  F  E. 

The  central  angle,  B  G  C,  of  the  6°  curve,  is  equal  to  180  — 
F  G  E  =  the  sum  of  the  angles  at  E  and  F  =  72°  12'  -f-  19° 
28'  =  91°  40',  making  the  arc  B  C  =  1,528  feet.  The  arc  B  A, 
equivalent  to  19°  28'  of  a  1°  curve,  =  1,947  feet.  Points  C  and 
A  being  thus  ascertained,  curve  A  C  may  be  located.  It  will 
consume  an  angle  =  180°  —  72°  12'  =  107°  48',  equivalent,  on 
an  8°  curve,  to  1,347.5  feet. 


XXXV. 

TO    LOCATE    A    TANGENT  TO  A  CURVE  FROM  AN 
OUTSIDE  FIXED  POINT. 

1.  If  the  ground  is  open,  and  the  curve  can  be  seen  from  the 
fixed  point,  it  may  be  marked  by  stakes  or  poles  at  short  inter- 
vals, and  the  tangent  laid  off  without  more  ado. 

2.  Suppose,  however,  that  on  cumbered  ground  a  trial  tan- 
gent, A  B,  has  been  run  out,  intersecting  the  curve  at  B:  it  is 
required  then  to  find  the  angle  BAE,  in  order  that  the  true 
tangent  A  E  may  be  laid  down. 


108  TO   SUBSTITUTE  A    CURVE. 


Example. 

A  B  =  1,500  feet ;  D  H  B,  a  4°  curve ;  angle  F  B  D  =  20°  13'. 

First,  the  angle  FED,  between  a  tangent  and  a  chord,  is 
equal  to  half  the  central  angle  subtended  by  the  same  chord. 
Angle  D  C  B,  therefore,  =  40°  26'.  By  Table  XVI.,  the  chord 
of  40°  26',  for  a  1°  curve,  =  3,960.2  feet;  for  a  4°  curve,  it  is, 
say,  990  feet  =  D  B ;  and  D I  =  I B  =  495  feet.  The  versin. 
II I  is,  in  like  manner,  found  to  be  88.25  feet.  Deducting  this 
from  the  radius  of  the  4°  curve,  we  have  I C  =  1,344.4  feet. 

Then  IC-i-IA  =  tan.  I  AC;  i.e.,  1,344.4 -r-  (495  -f  1,500) 
=  0.674  =  tan.  33°  59'  =  angle  I A  C. 


Next,  by  geometry,  the  proposed  tangent  A  E  =\/A  D  X  A  B 
=  \/2,490  X  1,500  =  1,932.6 ;  and  E  C  -f-  A  E  =  tan.  E  A  C  = 
1,432.69-:- 1,932.6  =  0.7413  =  tan.  36°  33'  =  angle  E  A  C.  Then 
E  A  C  —  I  A  C  =  36°  33'  —  33°  59'  =  2°  34'  =  angle  B  A  E,  the 
angle  required,  which  can  accordingly  be  laid  off  from  the  fixed 
point  A,  and  the  tangent  located. 


XXXVI. 

TO  SUBSTITUTE  A  CURVE  OF  GIVEN  RADIUS  FOR 
A  TANGENT  CONNECTING  TWO  CURVES. 

Example. 

1.  A  B,  a  4°  curve;  BC  =  774  feet;  CD,  a  6°  curve:  to  put 
in  the  1°  curve,  E  F. 

Sketch  the  figure  as  in  margin,  HK  being  parallel  and  equal 
to  BC.  Then  KG  =  BG  —  BK  or  CH  =  1,433  —  955  = 
478  feet;  KH  -7-  GK  =  774  H-  478  =  1.62  =  tan.  58°  19'  = 
angle  KGH;  and  KIT  -r-  sin.  58°  19'  =  774  -f-  0.851  =  909.6 
feet  =  G  H. 


TO  RUN  A    TANGENT  TO   TWO   CURVES. 


109 


In  the  triangle  GUI  we  have  then  the  sides  given;  namely 
G  H  =  say,  910  feet,  II I  =  5,730  —  955  =  4,775  feet,  and  GI 
=  5,730  —  1,433  =  4,297  feet :  to 
find  the  angles. 

Under  Case   3,  Trigonometry 

(in.),  ill  :  IG  +  GH  ::  IG 

—  G  H  :  I  L  —  L II ;  i.e.,  4,775  : 
5,207  : :  3,387  :  3,093,  the  differ- 
ence of  the  segments  into  which 
the  base  III  is  divided  by  a 
perpendicular  from  G.  Adding 
half  the  difference  of  the  seg- 
ments thus  found  to  half  their  sum,  the  longer  segment,  I  L,  is 
found  to  be  4,234  feet;  subtracting  half  the  difference  from 
half  the  sum,  the  shorter  segment,  LH,  is  found  to  be  541  feet. 
Then  H  L  -r-  II G  =  541  ^  910  =  0.5945  =  cos.  53°  31'  = 
angle  GUI.  In  like  manner,  dividing  IL  by  IG,  we  find  the 
angle  G I II  to  be  9°  49'.  The  sum  of  these  angles  =  angle 
E  G II  =  G3°  20',  for  the  reason  that  each  is  equal  to  180  — 
II G  I.  Finally,  E  G  H  —  K  G  H  =  63°  20'  —  58°  19'  =  5°  01' 
=  angle  E  G  B,  equivalent  to  a  distance  from  B  of  125  feet 
around  the  4°  curve  to  the  P.  C.  C.  at  E;  and  GIH  — EGB 
=  9°  49'  —  5°  01'  =  4°  48'  =  angle  CHF,  equivalent  to  a 
distance  from  C  of  80  feet  around  the  6°  curve  to  the  P.  C.  C. 
atF. 


XXXVII. 

TO  RUN  A  TANGENT  TO  TWO  CURVES    ALREADY 
LOCATED. 

1.  If  one  curve  be  visible 
from  the  other,  or  if  both 
be  visible  from  some  inter- 
M  mediate  point,  mark  them 
on  the  ground  with  stakes 
at  short  intervals.  The 
points  M  or  L  in  the  range 
of  the  required  tangent  may 

then  be  fixed  by  one  or  two  trial  settings  of  the  transit,  and 

the  line  put  in. 


110  TO  RUN  A    TANGENT   TO   TWO   CURVES. 

2.  Should  obstacles  prohibit   this   plan,   measure   any  con- 
venient line,  FG  or  BCD,  from  one  to  the  other  curve,  and, 
completing  the   traverse  A  F  G  E   or  A  13  C  D  E,   determine 
thence  the  bearings  and  distances  asunder  of  the   centres  A 
and  E.  '  The  right  triangle  A  E  K,  in  which  E  K  =  the  sum  of 
the  radii,  may  then  be  solved,  and  the  points  II  and  I  ascer- 
tained as  in  the  following  example:  — 

Example. 

F  B,  a  4°  curve;  G  D,  a-G°  curve.     X.  S.          E.  W. 

A  B,  N.  20°  E.,  1,433  feet     .     .     1,340.6  490.0 

B  C,  East,          3,570  feet      .     .  3,570.0 

CD,  X.  34°  E.,  1,800  feet      .     .     1,492.2  1,006.2 

DE,  X.  45°  W.,    955  feet     .     .        675.2  675.2 

3,514.0  5,066.2      675.2 

Total  northing,  3,514  feet;  total  easting,  4,391  feet. 

Then  4,391  -$-  3,514  =  1.2496  =  tan.  50°  20'  =-  bearing 
AE;  and  4,391  -f-  sin.  50°  20'  =  5,704  feet  =  distance  AE. 
Also,  EK  -^-  AE  =  (1,433  -f  955)  -!-  5,704  =  xin.  24°  45'  = 
angle  E  AK;  and  angle  AEK  =  90°  00'  —  24°  45'  =  65°  10'. 
Hence  the  bearing  of  AK  or  III  is  X.  75°  05'  E.,  and  that  of 
AHorIE,N.  14°  55' W. 

Since  AB  bears  X.  20°  E.,  the  angle  II AB  =  20°  00'  -f  14° 
55'  =  340  55^  equivalent  to  a  distance  of  873  feet  from  B  around 
the  4°  curve  to  the  required  P.  T.  at  II;  and,  since  DE  bears 
N.  45°  00'  W.,  the  angle  IE  D  =  45°  00'  —  14°  55'  =  30°  05', 
equivalent  to  a  distance  of  501  feet  from  D  around  the  6°  curve 
to  the  required  P".  C.  at  I. 

3.  Should  the  curves  turn  in  the  same  direction,  the  side 
E  K  of  the  triangle  A  E  K  is  equal  to   the   difference  of  the 
radii  instead   of  their  sum.     In   other  respects,   the  method 
exemplified  will  apply  to  that  case  also. 

4.  The  preceding  solution   may  be  useful   as  an  exercise. 
But  the  problem  is  one  of  rare  occurrence,  and  the  conditions 
must  be  extraordinary  which  prevent  a  close  approximation, 
at  least,  to  the  true  line  in  the  field.     The  better  way  in  actual 
practice,  then,  is  to  run  out  a  trial  tangent  as  nearly  right  as 
possible.     If  it  errs  by  passing  outside  the   objective  curve, 
close  with  a  compound  (XXIX.);  if  that  error  be  inadmissible, 
or  if  it  errs  by  cutting  the  objective  curve,  measure  the  miss, 
and  divide  it  by  the  length  of  the  trial  tangent.     The  quotient 


TO  RUN  A    TANGENT   TO   TWO   CURVES,  111 

will  be  the  natural  tangent  of  the  angle  of  retreat  or  advance 
on  the  first  curve  required  to  make  the  tangent  fit. 

5.  A  still  closer  adjustment  would  be,  after  determining  the 
angle  approximately  as  above,  to  find  the  "tangents"  corre- 
sponding to  it  for  the  two  curves  in  Table  XVI.  Subtract  the 
sum  of  these  tangents  from  the  length  of  the  trial  line,  if  it 
cuts  the  objective  curve;  add  the  sum,  if  it  passes  outside. 
With  the  number  thus  found,  divide  the  measured  amount  of 
error  for  the  tangent  of  the  angle  of  retreat  or  advance,  as  the 
case  may  be. 

G.  Suppose,  for  illustration,  that  a  trial  tangent,  bearing  by 
needle  X.  54°  30'  E.,  is  run  out  from  stake  24.80  of  a  4°  curve, 
intending  to  touch  a  G°,  but  is  found  to  cut  it.  Suppose  fur- 
ther that  the  objective  0°  curve  was  laid  down  and  numbered 
in  the  direction  of  approach  towards  the  4°  curve;  that  its  P. 
C.  is  stake  25.10,  and  the  magnetic  bearing  of  its  initial  tan- 
gent S.  30°  30'  \V.  The  angle,  then,  between  the  bearing  of  the 
trial  tangent  and  that  of  the  initial  tangent  of  the  G°  curve,  is 
24°,  corresponding  to  a  distance  of  400  feet  on  the  latter  curve. 
At  stake  -25.10  +  4.0  =  20.10,  therefore,  a  tangent  to  the  G° 
curve  would  be  parallel  to  the  trial  tangent.  Go  forward  on 
the  trial  tangent,  accordingly,  to  a  point  opposite  29.10,  and 
measure  the  distance  square  across  to  that  plus  on  the  6° 
curve.  Assuming  the  trial  tangent  to  be  2,500  feet  long,  and 
the  amount  of  the  miss  to  be  87  feet,  the  nat.  tan.  of  the 
angle  of  error  is  0.0348  =  tan.  2°.  By  the  method  in  (4),  this 
calls  for  a  shift  of  the  P.  T.  50  feet  ahead  on  the  4°  curve, 
making  the  new  P.  T.  24.80  -f  0.50  =  stake  25.30,  and  ad- 
vances the  P.  T.  of  the  G°  curve  to  stake  29.43  of  that  numera- 
tion. 

The  method  in  (5),  applied  to  this  case,  brings  the  angle  of 
error  2°  02',  instead  of  2°,  equivalent  to- a  deviation  of  1£  feet 
scant  in  half  a  mile  from  the  line  corrected  by  the  method  in 
(4),  and  agreeing  exactly  with  the  correction  determined  by 
the  method  in  (2). 


TRACK    PROBLEMS. 
XXX  VIII. -LI. 


TRACK    PROBLEMS. 


XXXVIII. 

REVERSED  CURVES. 

The  following  problems  will  be  useful  in  laying  off  turnouts, 
the  adjustment  of  tracks  near  stations  or  shops,  and  the  like; 
but  reversed  curves  should  never  be  used  on  the  main  line 
between  stations,  where  they  are  both  objectionable  and  unne- 
cessary. Ground  which  allows  any  permissible  location  at  all 
will  allow  straight  reaches  of  at  least  two  hundred  or  three 
hundred  feet  between  curves  of  contrary  flexure;  and  in  every 
case  it  is  worth  the  small  additional  outlay  to  make  such  a 
location. 


XXXIX. 

TO    CONNECT    TWO    PARALLEL    TANGENTS    BY    A 
REVERSED  CURVE  HAVING  EQUAL  RADII. 

1.  The  radius  R,  and  the  perpendicular  distance  D,  between 


110  TO   CONNECT  TWO  PARALLEL   TANGENTS. 

Draw  the  tangents,  radii,  and  curves,  fixing  the  P.  R.  C. 
midway  of  D. 

Draw  the  chords  G  I,  I E,  the  line  B  F  perpendicular  to  G  I, 
and  the  line  E  H  in  prolongation  of  radius  CE  to  an  intersec- 
tion with  B  H  passed  through  centre  B  parallel  to  tangents. 

That  1  falls  midway  of  D,  follows  from  the  necessary  sym- 
metry of  the  figure;  and  G  IE  must  be  a  straight  line,  because 
the  radii  B  I,  1C,  perpendicular  to  a  common  tangent  at  the 
same  point,  form  a  straight  line,  to  which  the  chords  G  I,  IE, 
are  equally  inclined. 

C  H-^  C  B  =  cos.  A;  but  C  H  =  2  R  —  D,  and  C  B  =  2  R. 

.-.  cos.  A  =  (2  II  —  D)-|-2R. 

B  H  =  B  C  sin.  A  =  2  R  sin.  A ; 

GF  =  R«Mi.  |  A;  GE  =  4GF. 

.  •.  G  E  =  4  R  sin.  $  A,  and  GIorIE  =  2R  sin.  I  A. 

Observe,  that,  in  the  right  triangles  G  K  E  and  B  G  F,  the 
angles  at  G  and  B  are  each  equal  to  |  A:  hence  the  triangles 
are  similar. 

Example. 

R  =  800  feet,  D  =  24  feet. 
To  find  angle  A. 

Cos.  A  =  (2  R  —  D)  -r-  2  R  =  1,576  -f-  1,600  =  0.985  =  nat. 
cos.  9°  56'. 

BH  may  then  be  found  =.2  R  sin.  A  =  1,600  X  0.1725  = 
276  feet,  and  laid 'off  from  the  P.  C.  at  G  to  K,  the  point  E 
being  fixed  by  a  right  angle  from  K. 

Or  GE  may  be  found  =  4  R  sin.  i  A  =  3,200  X  0.866  = 
277.1  feet,  and  laid  off  from  G  to  E,  the  point  I  being  fixed 
138.5  feet  from  G,  and  angle  K  GE  made  equal  to  half  of  A  = 
4°  58'. 

2.  The  distances  G  K  and  D  given,  to  find  R. 

In  triangle  GKE,  KE  =  D. 

D  -T-  GK  =  tan.  £  A;  D  -j-  sin.  i  A  =  GE;  and  GE  -f- 
sin.  $  A  =  4  R. 

Or,  having  found  GE,  we  have  from  the  congruity  of  trian- 
gles GKE,  BEG, 

D  :  GE  ::  i  GEorGF  :  R. 


TO   CONNECT  TWO  PARALLEL    TANGENTS. 


117 


Example. 
=  300  feet,  D  =  28  feet. 


D-f-GK     .     .    .    . 

=  Tan.  \ 
D  -r-  sin.  $  A   .    .     . 

=  GE 

G  E  -f-  .sin.  -J-  A    .     . 


.     Log.    28 
Log.  300 

A    .     5°  20' 

.     Log.    28 
Sin.  5°  20' 

.     .    301.24 

.  Sin.  5°  20' 


=  4  R  .  .   3,241 
.'.  R  =  810.2. 


1.447158 
2.477121 

8.970037 

1.447158 
8.968249 

2.478909 
8.968249 

3.510660 


XL. 


TO  CONNECT  TWO  PARALLEL  TANGENTS  BY  A 
REVERSED  CURVE  HAVING  UNEQUAL  RADII. 


1.  Given  the  perpendicular  distance,  D,  between  two  paral- 
lel tangents,  and  the  unequal  radii,  R  and  r,  of  a  reversed 
curve,  to  find  the  central  angles,  A,  the  chords,  and  the 
straight  reach,  G  K,  of  the  curve. 


118  TO   CONNECT  TWO  PARALLEL   TANGENTS. 

Cos.   A  =  C  H  -f-  B  C  ;    but   C  H  =  (R  +  r)  —  D,    and 

B  C  =  K  +  r. 
.'.  Cos.  A=  (K'-fr  —  D)H-  (R  -f  r). 


The  straight  reach  GK  =  BH=(K-|-r)  sin.  A. 
The  sum  of  the  chords  G  E  =  G  K  -=-  cos.  |  A. 

GI  =  2Rsm.  i  A. 
IE  =  2  r  sin.  i  A  =  GE  —  GI. 

Example. 
D  =  28,  E  =  955,  r  =  574. 

Cos.  A  =  (R  -f  r  —  D)  —  (R  +  r)  =  1,501  ~  1,529. 

1,501  .  .  .  log.  3.176381 
1,529  .  .  .  log.  3.184407 

Cos.  A,  10°  59'  .....  9.991974 

GK=  (R  +  r)  sin,  A. 

R  +  r,  1,529  .  .  .  log.  3.184407 
Sin.  A,  10°  59'  ...  log.  9.279948 

GK  =  291.3  .....  2.464355 

G  E  =  G  K  —  cos.  -J  A. 

GK,  291.3  ...  log.  2.464355 
Cos.  i  A,  5°  29i'  .  .  .  log.  9.998014 

GE  =  292.6  .....  2.466341 


2  R,  1,910    .     .     .    log.  3".281033 
Sin.  i  A,  5°  29^    ...     log.  8.980916 

G  I  =  182.8    .....  2.261949 
IE  =  GE  —  GI  =  292.6  —  182.8  =  109.8. 

2.  The  distances  GK  and  D,  and  one  of  the  unequal 
radii,  R,  given,  to  find  the  other  radius,  r,  and  the  central 
angles,  A. 


REVERSED   CURVE  WITH  UNEQUAL  ANGLES.  119 

Example. 
G  K  =  422,  D  =  30,  R  =  2,292. 

Tan.  |  A  =  D  -f-  G  K. 

D  =  30  .  .  .  log.  1.477121 
GK  =  422  .  .  .  log.  2.625312 

Tan.  i  A,  4°  04'  .  .  .  .  .  8.851809 

.-.  A  =  8°  08'. 

G  E  =  D  -r-  sin.  i  A. 

D  =  30    .     .     .     log.  1.477121 
Sin.  i  A,  4°  04'    ...     log.  8.850751 

GE  =  423 2.626370 

GI  =  2Rsw.  i  A. 

2  R  =  4,584  ...  log.  3.661245 
Sin.  |  A,  4°  04'  .  .  .  log.  8.850751 

GI  =  325.1 2.511996 

GE  —  G  I  =  423  —  325  =  98  =  IE. 

r  =  $<L  E  -r-  sin.  $  A. 

iIE  =  49  .  .  .  log.  1.690196 
Sin.  I  A,  4°  04'  .  .  .  log.  8.850751 

r  =  691          .  2.839445 


XLI. 

A  REVERSED  CURVE  HAVING  UNEQUAL  ANGLES. 

Given  the  angles  A  and  B,  and  the  length  A  B  of  a  straight 
line  connecting  two  diverging  tangents,  to  find  theradius  of  a 
reversed  curve  to  close  the  angles. 

A I  =  R  X  tan.  i  A ;  B I  =  R  X  tan.  |  B. 
.-.  A  B  ==  R  X  (tan.  $  A  +  tan.  £  B). 
.-.  R  =  AB-f-  (tan.  $  A  -f  tan.  $  B). 


120          REVERSED   CURVE  BETWEEN  FIXED  POINTS. 

Example. 
A,=  16°,  B  =  10°,  A  B  =  840. 


A  B,  840     o log.  2.924279 

I  -A  =  8°,  nat.  tan.  0.14054 

£  B  =  5°,  nat.  tan.  0.08749  • 

Tan.  i  A  +  Ian.  }  B  =  0.22803     .    .      log.  —1.357992 

R  =  3,600 .  3.566287 


XLII. 

A  REVERSED  CURVE  BETWEEN  FIXED  POINTS. 

Given  the  angles  N  and  K,  and  the  length  of  the  straight 
line  E  F  connecting  two  divergent  tangents,  to  find  the  radius 
of  a  reversed  curve  from  E  to  F,  connecting  the  tangents. 

1.  Denote  the  angle  EIC  or  DIF  by  I;  the  angle  CEI, 
complement  of  N,  by  n ;  and  the  angle  D  F  I,  complement  of 
K,  by  k. 

Then,  in  triangle  E  C  I,  — 


E  c  :  c  I : :  sin.  I 


.  •.  E  C  X  sin.  n  =  C I  X  sin.  I. 


REVERSED   CURVE  BETWEEN  FIXED  POINTS. 


121 


Also,  in  triangle  D  F  I,  -— 
D  fi  :  D I  : :  sin.  I  :  sin.  k.    .'.DFX  sin.  k  =  D I  X  sin.  I. 

Adding  these  equations,  we  have  — 

EC  Xsin.  n-f  DF  X  sin.  fc  =  (CI-fDI)  X  sin.  I. 


But  EC  and  DF  are  each  equal  to  R;  sin.  n  =  cos.  N;  sin. 
k  =  cos.  K;  and  CI  +  DI  =  2  R. 
Hence  the  equation  becomes,  — 

R  X  (cos.  N  +  cos.  K)  =  2  R  X  sin.  I. 
.'.  sin.  I  =  (cos.  N  -f-  cos.  K)  -4-  2. 

The  foregoing  elegant  solution  is  abridged  from  Henck. 
2.  Angle  A  =  180  —  \n  -f  I) ;  angle  B  =  180  —  (  k  -f  I). 
To  find  radius,  draw  F  H  parallel,  and  E  H  perpendicular,  to 
CD. 

ThenEH  =  EF  Xsin.  I. 

But  EH  =  EG  +  GH;  EG  =  R  X  sin.  A;  andGH  =  R 
X  sin.  B. 

.'.  EF  X  sin.  I  =  R  X  (sin.  A  -f  sin.  B). 
.-.  R  =  E  F  X  sin.  I  4-  (sin.  A  -j-  sin.  B). 


122         REVERSED   CURVE  BETWEEN  FIXED  POINTS. 

Example. 
E  F  =  1,400,  N  =  30°,  K  =  20°.  * 

Sin.  I  =  (cos.  N  -f  cos.  K)  -f-  2. 

N  =  30°,  nat.  cos 0.86603 

K  =  20°,  nat.  cos 0.93969 

1.80572 

1.80572  -T-  2  =  0.90286  =  nat.  sin.  64°  32'.. 
. '.  I  =  64°  32'.    . 

A  =  180°  —  (n  -f- 1)  =  180°  —  (60°  +  64°  32')  =  55°  28'. 
B  =  180°  —  (k  +  I)  =  180°  —  (70°  +  04°  32')  =  45°  28'. 
R  =  E  F  X  sin.  I  -f-  (sin.  A  -f  sin.  B). 

EF  =  1,400    .     . log.  3.146128 

Nat.  sin.  I,  0.90286 .......       log.  —1.955621 

EG  =  1,264 3.101749 

A  =  55°  28'  nat.  sin 0.82380 

B  =  45°  28'  nat.  sin.       ,  0.71284 


Sin.  A -{-sin.  B 1.53664  log.  0.186579 

R  =  822.6 2.915170 

3.  The  young  student  should  bear  in  mind  that  the  addition 
or  subtraction  of  the  logarithms  of  two  natural  numbers  gives 
a  logarithm  representing,  not  the  sum  or  difference,  but  the 
product  or  quotient,  of  such  numbers.  When,  therefore,  as  in 
the  two  foregoing  cases,  the  sum  or  difference  of  two  or  more 
trigonometric  functions  —  sines,  tangents,  and  the  like  — is 
sought,  the  logarithm  of  the  sum  of  the  natural  functions,  and 
not  the  sum  of  their  logarithms,  is  to  be  used.  If,  for  example, 
sin.  A  X  sin.  B  is  required,  the  log.  sin.  A  -f-  log-  sin.  B  =  the 
logarithm-  of  the  product  of  the  sines  designated ;  but,  if  sin.  A 
-f-  sin.  B  is  sought,  the  natural  sines  of  those  angles  must  be 
added  together,  and  the  logarithm  of  the  sum  of  these  natural 
functions  must  be  used  in  making  logarithmic  calculations. 


TO   CONNECT  TWO  DIVERGENT   TANGENTS. 


123 


XLIII. 

TO    CONNECT    TWO    DIVERGENT   TANGENTS  BY  A 
REVERSED  CURVE. 


1.    ADVANCING  TOWARDS  THE    INTERSECTION   OF    TANGENTS. 

Given  the  angle  of  divergence,  N,  the  initial  P.  C.  at  G, 
the  distance  GH,  and  the  radii  R,  r,  to  find  the  central  angles 
A  and  B. 


GK  =  CG  X  cos.  N  = 
GL  =  GH  X  Kin.  N. 

GK  —  GL  =  LKorEF,  CF  being  drawn  parallel  to  L  E. 
Cos.  B  =  DF-^DC  =  (r  +  EF)  -f-  (R+r). 
Angle  GC  K  =  90°  —  N  ;  angle  D  C  F  =  90°  —  B. 
AngleA  =  GCK  —  DCF  =  (90°  —  N)  —  (90°  —  B)  =  B 
N. 

Example. 
N  =  24°  30',  G  H  =  854,  R  =  1,440,  r  =  1,146. 


G  K  =  R  cos.  N. 


R  =  1,440 

Cos.  N,  24°  30' 

GK  =  1,310 


log.  3.158362 
log.  9.959023 

.  3.117385 


124  TO   CONNECT  TWO  DIVERGENT  TANGENTS. 

GL  =  GH  Xsm.  N. 

G II  =  854    .     .     .     log.  2.931458 
Sin.  N,  24°  30'    ...     log.  9.617727 

G  L  =  354 2.549185 

LK  or  E  F  =  GK  —  G  L  =  1,310  —  354  =  95G. 
Cos.  B  =  (r  -f  E  F)  -7-  (R  +  r). 


=  2,102    . 

=  2,586    . 

Cos.  B,  35°  38'    . 


log.  3.412629 
.  9.910004 


B  =  35°  38'. 

A  =  B  —  N  =  35°  38'  —  24°  30'  =  11°  08'. 


X" 


JF 


2.    RECEDING   FROM  THE   INTERSECTION  OF   TANGENTS. 

Given  the  angle  of  divergence,  N,  the  initial  P.  C.  at  G, 
the  distance  G  H,  and  the  radii  K,  r,  to  find  the  central  angles 
A  and  B. 

GK  =  GH  X  tan.  N. 
KC  =  GC  — GK=-R  — GK. 

L  C  or  E  F  =  K  C  X  cos.  N,  the  line  C  F  being  drawn  paral- 
lel to  L  E. 

Cos.  B  =  D  F  -=-  C  D  =  (r  -f  E  F)  -7-  (B  +  r). 
Angle  A  manifestly  =  B  +  N". 


TO  SHIFT  A   P.  R.    C.  125 


Example. 
\  =  18°  30',  G  H  =  920,  R  =  955,  r  =  819. 

GK  =  GH  X  tan.  N. 


=  920    .     .     .     log.  2.963788 
Tan.  18°  30'    ...     log,  9.524520 

GK  =  307.8 2.488308 

K  C  =  R  —  G  K  =  955  —  307.8  =  647.2. 
LCorEF  =  KC  X  cos.  N. 

KC  =  647.2    .     .    .    log.  2.811039 
Cos.  N,  18°  30'    .     .     .    log.  9.976957 

EF  =  613.8    .     .  ^  .    .    .  2.787996 

Cos.  B  =  (r  4-  E  F)  -MR  +  r). 

r  4-  E  F  =  1,432.8  .  .  .  log.  3.156185 
R  4-  r  =  1,774  .  .  .  jog.  3.248954 

Cos.  B,  36°  08' 9.907231 

B  =  36°  08 

A  =  B  4-  N  =  36°  OS'  4- 18°  30'  =  54°  38'. 


XLIV. 

TO  SHIFT  A  P.  R.  C.  SO  THAT  THE  TERMINAL 
TANGENT  SHALL  MERGE  IN  A  GIVEN  TANGENT 
PARALLEL  THERETO. 

Given  the  reversed  curve  EFG,  ending  m  tangent  GV:  to 
find  the  angle  of  retreat,  A,  on  the  first  branch,  and  the  angle 
C  of  the  second  branch,  ending  in  tangent  II T,  parallel  to 
GV. 

Measure  the  error  T  G  =  D,  perpendicular  to  the  terminal 
tangent. 


126 


TO  SHIFT  A   P.   R.    C. 


In  the  figure,  draw  L  K  parallel  to  G  V,  and  passing  through 
centre  of  first  branch. 


Then  M  K  =  (R  +  r)  X  cos.  B. 

NL  =  (R  +  r)  X  cos.  C. 

WL  =  GK. 

^TL  =  r-f-D-f  GK. 

M  K  =  r  -f  G  K. 

NL  —  MK  =  D. 

.-.  (R  +  r)  X  cos.  C  —  (R  -4-  r)  X  cos.  B  =  D. 

.-.  (R  +  r)  X  cos.  C  =  (R  +  r)  X  cos.  B  +  D. 

.-.  Cos.  C  =  [(R  +  r)  X  cos.  B  +  D]  -7-  (R  +  r). 

A  =  (90°  -  C)  -  (90°  -  B)  =  B  -  C. 

Example. 

R  =  1,433,  r  =  819,  B  =  34°  20',  D  =  94. 
Cos.  C  =  [(R  +  r)  cos.  B  +  D]  -r-  (R  +  r). 

R-f  r  =  2,252    .     .     .     log.  3.352568 
B  =  34°  20',  cos.     .  -  .     .     log.  9.916859 

(R  -f  r)  cos.  B  =  1,860    ....     .3.269427 
Add  D  94 


1,954 

(R  +  r) 


log.  3.290925 
log.  3.352568 


Cos.  C,  29°  49' 9.938357 

A  =  B  —  0  =  34°  20'  —  29°  49'  =  4°  31'. 


CURVE  THROUGH  A  FIXED  POINT. 


127 


XLV. 

TO    PASS    A    CURVE    THROUGH    A    FIXED    POINT, 
THE  ANGLE  OF  INTERSECTION    BEING  GIVEN. 


Given  the  Intersection  angle,  A,  of  two  tangents,  to  find  the 
radius,  R,  of  a  curve  which  shall  pass  through  a  point,  C; 
the  position  of  said  point,  with  reference  to  the  tangents  or  the 
point  of  intersection,  being  known. 

1.  By  what  data  soever  point  C  is  located,  they  may  be  com- 
muted by  simple  processes  to  the  form  shown  in  the  figure; 
namely,  the  ordinate  BC  and  the  distance  1C  to  apex.  Call 
the  angle  B  I  C  a,  and  complete  the  triangle  ICO. 

In  this  triangle,  x  =  /180~A\  —  a  =  00°  —  U  A  +  a). 
Also,  C  O  :  I  O  :  :  sin.  x  :  sin.  z. 

But  C  O  =  R  ;  I  O  =  —  —-.      .  '.  R  :      Rr  .   :  :  sin.  x  :  sin.  z. 
cos.  $  A  cos.  i  A 


Hence  sin.  z 


' 


A  .     The  triangle  ICO  may  then  be 

.        J\- 

solved,  and  the  radius  ascertained. 


128 


CURVE   THROUGH  A   FIXED  POINT. 


Example. 

A  ==  40°,  B  C  =  32  feet,  I  B  =  80  feet. 
Then  BC  -4-  IB  =  32  -f-  80  =  0.4  nat.  tan.  21°  49';   and 
I  C  =  B  C  -f-  nat.  sin.  21°  49'  =  32  +  .372  =  86  feet. 
Also,  x  =  90°  —  (|  A  +  a)  =  90°  —  (41°  49')  =  48°  11'. 


Next,  sin.  x,  48°  11' 
Divided  by  cos.  }  A,  20° 

=  sin.  *,  127°  31' 


log.  9.872321 
lo.  9.972986 


log.  9.899335 


Or,  since  the  sine  of  any  angle  is  equal  to  the  sine  of  its  sup- 
plement, the  supplement  in  this  case,  52°  29',  may  be  taken 
directly  from  the  logarithmic  table,  from  which  supplement 
deducting  x,  or  48°  11',  the  remainder  is  the  angle  y  =  4°  18'. 


Finally,  1C  =  86 
Multiplied  by  sin.  x,  48°  11' 

=  CD 
And  C  D  divided  by  sin.  y,  4°  18' 

=  C  O  =  K  =  say,  855  feet 


log.  1.934498 
log.  9.872321 

log.  1.806819 
log.  8.874938 

lo.  2.931881 


2.  In  the  case  of  a  rectangular  intersection,  the  solution  is 
more  simple.     It  is  quite  plain,  from  the  figure,  that  — 


from  which  equation, 

R  =  a  -f- 


«i 


FROGS  AND  SWITCHES. 


129 


Example. 

a  =  40,  6  =  80. 

Then  R  =  40  -f  80  -f  V6,400  =  200. 

3.  Cases  of  this  kind  are  disposed  of  with  great  ease  in  the 
field  by  means  of  the  curve-protractor. 


XLVI. 
FROGS  AND  SWITCHES. 


TO  FIND  THE  RADIUS  OP  A  TURNOUT  CURVE,  THE  FROG 
ANGLES,  AND  THE  DISTANCES  FROM  THE -TOE  OF  SWITCH 
TO  THE  FROG  POINTS. 

1.  Draw  the  figure  as  in  margin,  C  being  the  centre  of  the 
turnout  curve,  C  K  parallel  to  main  track,  and  O  K,  I E,  L  M, 
perpendicular  to  it.     Call  the  angle  of  the  frogs  at  O,  F;  that 
of  the  intermediate  frog  at  I,  2  F';  the  throw  of  the  switch -rail 
for  single  turnout,  D;  its  angle  with  main  track,  S;  the  gauge 
of  the  track,  G;  and  radius  of  outer  rail,  R. 

2.  Usually  the  length  and   throw  of    switch-rail   and    the 
angles  of  the  frogs  at  O  are  given.     In  that  case,  to  find  R,  F7, 
and  the  distances  LO,  LI,  reason  thus:  — 


130  FROGS  AND  SWITCHES. 

3.  The  angle  UN  W,  between  the  line  of  the  switch-rail  pro 
longed   and   a  tangent   to  turnout  curve  at  frog  point  O,  = 
NOP  —  NIIW  =  F  —  S.     The  angle  NOL  or  NLO,  be- 
tween chord  and  tangent,  =  half  the  intersection  angle  UN  W 
=  i  (F  —  S).    The  angle  NOB  =  NOL-f-LOB.    But  N  O  L 
is  seen   to  be  =  4  (F  —  S),  and  NOB  =  F;  then  LOB  = 
NOB  —  NOL  =  F  —  $  (F  — S)=|  (F-j-S).     The  distance 
LO,  from  toe  of  switch  to  point  of  main  frog,  =  LB  -f-  sin. 
LOB  =  (G  —  Dj  -7-  sin.  |  (F  -f  S). 

4.  Again:  the  .angle  LCY  =  NLO  =£  (F  — S);  LY  =  | 
LO  =  i  (G  —  D)  -f-  sin.  $  (F  +  S).     L  Y  -f-  sin.  LC  Y  = 
LC;  i.e.,  [|  (G  —  D)  -=-  sin.  i  (F  -f-  S)]  -f-  sin.  i  (F  —  S) 
=  R. 

5.  R  may  be  found  otherwise,  as  follows:  — 

O  K  =  6  C  co.s.  K  O  C  =  R  cos.  F ;  L  M  =  L  C  co.s.  C  L  M  = 
R  cos.  S;  LM  —  OK  =  LB;  i.e.,  R  (co.s.  S  —  co.s.  F)  =  (G  - 
D).  Hence  R  =  (G  —  D)  -i-  (nat.  co.s.  S  —  nat.  co.s.  F). 

6.  If  R  be  known,  to  find  F.     This  equation  gives  nat.  cos.  F 
=  nat.  co.s.  S  —  |(G  —  D)  -f-  R]. 

7.  To  find  the  angle,  2  F',  of  the  middle  frog  at  J. 

IE  =  IP  -j-  P  E  or  O  K;  i.e.,  R  cos.  F'  =  i  G  -f  R  co.s.  F. 
Hence  nat.  cos.  F'  =  nat.  co.s.  F  -j-  ($  G  -f-  R). 

8.  The  angle  L  I  V,  by  similar  reasoning  to  that  used  in  rela- 
tion to  LOB,  is  found  to  be  =  i  (F'  -f  S).     The  distance  LI, 
from  toe  of  switch  to  point  of  middle  frog,  =  L  V  -j-  sin.  LI  Y 
=  (|  G  —  D)  -f-  sin.  J  (F'  -f  S). 

The  preceding  formulas  translate  into  the  following  — 

HULKS   FOIl  FROGS   AND   SWITCHES. 

9.  To  find  the  Anyle  of  Switch-Rail  with  Main  Track. 
Divide  its  throw,  in  decimals,  by  its  length:  the  quotient 
will  be  the  natural  sine  of  the  angle  sought. 

10.  To  find  the  Distance  from  Toe  of  Switch  to  Point  of 
Main  Frog. 

Subtract  the  throw  of  switch-rail  from  the  gauge  of  track, 
both  in  decimals;  call  the  remainder  a.  Add  together  the 
awgle  of  switch-rail  with  main  track  and  the  angle  of  the 
main  frog,  find  the  natural  sine  of  half  this  sum,  and  call 
it  b.  Divide  a  by  b:  the  quotient  will  be  the  distance 
sought. 


FROGS  AND  SWITCHES.  131 

11.  To  find  the  Radius  of  Outer  Hail  of  Turnout  Curve. 
Subtract  the  throw  of  switch-rail  from  the  gauge  of  track, 
both  in  decimals;  call  the  remainder  a.  Subtract  the  natural 
cosine  of  the  main  frog  angle  from  the  natural  cosine  of  the 
switch-rail  angle;  call  the  remainder  b.  Divide  a  by  b:  the 
quotient  will  be  radius. 

12.  To  find  the  Main  Frog  Angle,  the  liadius  of  the  Outer 

Rail  being  known. 

Call  the  natural  cosine  of  the  switch-rail  angle. a.  Subtract 
the  throw  of  switch-rail  from  the  gauge  of  track,  both  in  deci- 
mals. Divide  the  remainder  by  radius;  call  the  quotient  b. 
Subtract  b  from  a:  the  remainder  will  be  the  natural  cosine  of 
the  main  frog  angle. 

13.  To  find  the  Angle  of  the  Middle  Frog,  in  the  Case  of 

a  Double  Turnout. 

Call  the  natural  cosine  of  the  main  frog  angle  a.  Divide 
half  the  gauge  of  track  by  the  radius  of  outer  rail  of  turnout 
curve;  call  the  quotient  b.  Add  a  and  b  together.  Their  sum 
is  the  natural  cosine  of  half  the  middle  frog  angle. 

14.  To  find  the  Distance  from  Toe  of  Sivitch  to  Point  of 

Middle  Frog. 

Subtract  the  throw  of  switch-rail  from  half  the  gauge  of 
track,  both  in  decimals;  call  the  remainder  a.  Add  together 
the  switch-rail  angle  and  half  the  middle  frog  angle.  Find. the 
natural  sine  of  half  this  sum;  call  said  natural  sine  6.  Divide 
a  by  b :  the  quotient  will  be  the  distance  sought. 

15.  The  use  of  logarithms  will  be  found  convenient  in  work- 
ing these  rules. 

Examples. 

16.  Switch-rail,  18  feet;  throw,  5  inches  =  0.42  feet";  frog 
angle,  5°  44';  gauge,  4.71  feet. 

Sin.  S  =  0.42  -4-18  =  .02334  =  sin.  1°  20'. 

LO  =  (G  —  D)  -f-  sin.  I  (F  -4-  S)  =  (4.71  —  0.42)  -=-  sin. 
3°  32'  =  4.29  -^  0.0616  =  69.64  feet. 

R  =  (G  —  D)  -r-  (nat.  cos.  S  —  nat.  cos.  F)  =  4.29 -^  0.00473 
=  907  feet. 

Nat.  cos.  F'  =  nat.  cos.  F  +  (£  G  -=-  R)  =  0.995  +  (2.354  -f- 


l:J2  FROGS  AND  SWITCHES. 

907)  =  0.99759  =  cos.  3°  58  J'.     Hence  the  angle  of  the  middle 
frog  =  2  F'  =  7°  57'. 

L I  =  (i  G  —  D)  -f-  sin.  |  (F7  +  S)  =  (2.354  —  0.42)  -f-  sin. 
i  (3°  581'  +  1°  20')  =  1-934  4-  0.0403  =  41.8  feet. 

17.  In  ordinary  practice,  frogs  may  be  located  with  sufficient 
exactness  by  the  following  rules,  deduced  from  the  congruity 
of  triangles.     Great  nicety  in  their  location  is  not  necessary. 
The  important  thing  in  practice  is  to  lay  the  turnout  curve  so 
that  the  approach  to  the  frog  shall  be  fair  and  regular.     How 
trackmen  may  do  this  without  the  use   of  instruments,  in  a 
very  simple  way,  will  be  shown  hereafter.     Not  that  frogs  may 
be  set  hap-hazard,    and   the   approaches   forced   to  .fit:   they 
ought  to  be  nearly  where  they  mathematically  belong,  and  they 
can  be  thus  placed  by  means  of  the  rules  subjoined. 

18.  Let  N  stand  for  the  number  of  the  frog; 

L  the  length  of  switch-rail  in  feet; 

F  the  distance  from  toe  of  outer  switch-rail  to  point 

of  frog  in  feet. 

Then,  for  standard  gauge,  4  feet  S£  inches,  straight  switch- 
rail,  and  5  inches  throw  of  switch. 

F  =      8.6  LN 

L  -f  0.42  N' 

The  above  may  be  written  roundly  as  a  rule  thus:  — 
Multiply  the  length  of  switch-rail  in  feet  by  the  number^of 
the  frog,  and  set  down  the  product.  Multiply  that  product  by 
8i,  and  call  the  result  A.  Next  "add  together  the  length  of 
SWKch-rai}  in  feet  and  two-fifths  of  the  frog  number;  call  the 
sum  B.  Then  divide  A  by  B,  and  the  quotient  will  be  the  dis- 
tance in  feet  from  toe  of  outer  switch-rail  to  point  of  frog. 

Example. 
Switch-rail,  20  feet  long;  frog,  No.  9. 

Length  of  switch-rail 20 

Multiplied  by  frog  number.     ....  9 

Product 180 

Multiplied  by 8$ 

1,530  =  A. 
Length  of  switch-rail    ......        20 

Added  to  I  frog  No.  9 3.6 

23.(5  —  B. 


FliOGS  AND   SWITCHES.  133 

A  divided  by  B  =  1,530  divided  by  23.6  =  64,8  feet,  the  frog 
distance;  say,  65  feet. 

19.  If  the  switch-rail  be  curved,  the  formula  would  stand 
thus :  — 

L  N 


•pi 


L  -f  0.84  N 


Which  may  be  made  a  written  rule  as  follows:  — 
Multiply  the  length  of  switch-rail  in  feet  by  the  number  of 
the  frog,  and  their  product  by  8£;  call  the  result  A.  Add 
together  the  length  of  switch-rail  in  feet  and  four-fifths  of  the 
frog  number;  call  the  sum  B.  Then  divide  A  by  B,  and 
the  quotient  will  be  the  distance  from  toe  of  outer  switch-rail 
\o  point  of  frog  in  feet. 

20.  The  foregoing  rules    are   applicable   to  turnouts    from 
curves,  as  well  as  from  straight  lines. 

21.  To  find  the  radius  of  outer  rail  of  a  turnout  curve  from 
straight  track.     Data  same  as  in  previous  rules  for  frogs;  R 
the  required  radius  in  feet. 

8.6  L2  X2 
If  the  switch-rail  be  straight,  R  =  r=—t  -  . 

8.6  L2  X2 
Jf  the  switch-rail  be  curved,  R  =  j  *  _  ^  ,.o  ^' 

22.  To  find  the  radius  of  the  outer  rail  of  a  turnout  curve 
from  curved  track,  proceed  thus:  — 

First  find  the  radius  as  for  a  turnout  from  straight  track  by 
the  preceding  rule;  call  it,  as  before,  R.  Call  the  radius  of  the 
main  track  R2,  and  the  required  radius  of  turnout  curve  r. 

Then,  if  the  turnout  be  towards  the  concave  side  of  main 
track,  — 

R.2  X  R 
~  R2  -f  R* 

If  the  turnout  be  towards  the  convex  side  of  main  track,  — 
R,  X  R 


More  explicitly,  in  the  first  case,  r  is  equal  to  the  product  of 
the  other  radii  divided  by  their  sum;  and,  in  the  second  case, 
r  is  equal  to  the  product  of  the  other  radii  divided  by  their 
difference. 


134          .  FROGS  AND  SWITCHES. 

23.  The  angle  of  a  frog  is  equal  to  3,440'  divided  by  the  frog 
number. 

24.  To  find  the  frog  distances  and  radii  for  a  three-foot 
gauge,  find  them  by  the  preceding  rules  for  standard  gauge, 
and  take  five-eighths  of  the  result,  using  a  switch-rail  reduced 
in  like  measure. 

For  a  metre  gauge,  take  seven- tenths  of  the  result,  using  a 
switch-rail  reduced  in  like  measure. 

Or  these  radii  and  distances  may  be  found  from  the  appended 
tables  for  standard  gauge  by  pro-rating  as  above.  .  • 

25.  Three   frog  patterns   are    enough    for  general    service. 
They  should  be  so  proportioned,  that,  taken  in  couples,  the  less 
may  fit  as  middle  frogs  on  double  turnouts.     Numbers  5£,  7£, 
and  10|  make  an  excellent  suit;  numbers  5,  7,  and  9J  also 
answer  very  well. 

*  26.  At  the  terminal  stations,  and  about  the  shops  of  busy 
roads,  patterns  necessarily  multiply.  The  better  way  in  such 
cases  is  to  plot  the  situation  to  a  large  scale,  and  to  take  the 
required  distances  and  angles  from  the  drawing. 


TURNOUT    TABLE. 


135 


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TURNOUT    TABLE. 


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frog  dist. 
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witch-rail, 
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Ang.4°00'. 
Rad.  171.9. 


dist. 
r  rail. 
dist. 
frog  angle 


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g  angle 


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TO  LOCATE  A    TURNOUT. 


137 


XL  VII. 

TO  LOCATE  A  TURNOUT. 

1.  Let  the  heavy  parallels  in  the  figure  represent  the  rails  of 
the  main  track. 


2.  Stick  a  pin  or  drive  a  spike  at  A,  the  toe  of  switch,  at 
a  distance  from  the  gauge  side  of  the  main-track  rail  equal  to 
the  throw  of  the  switch-rail.  Lay  off  the  distances  AC  and 
AB  (if  a  double  turnout),  taken  from  the  foregoing  tables,  and 
place  the  frogs  C  and  B,  or  mark  those  points.  Stretch  the 
cord  from  A  to  B,  and  from  B  to  C.  Mark  the  middle  points 
of  those  stretches  at  H  and  P.  Catch  the  cord  at  H  with  your 
forefinger,  and  pull  it  outwards  until  your  finger,  at  E,  lines 
with  the  switch-rail,  and  also  with  the  right  gauge  side  of  frog 
B.  Stick  a  pin  at  L,  half-way  between  H  and  E.  Let  the 
cord  spring  in  against  L,  so  that  it  shall  stretch  straight  from 
A  to  L,  and  from  L  to  B.  Opposite  the  middle  points,  V,  of 
those  stretches,  stick  pins  on  the  outside  at  a  distance  from 
the  cord  equal  to  one-quarter  of  H  L.  In  like  manner,  catch 
the  cord  at  P,  the  point  midway  between  B  and  0;  stretch  it 
to  F,  in  line  with  the  gauge  sides  of  the  frogs;  a,nd  stick  a  pin 
at  I,  half-way  between  P  and  F. 


138  TO  LOCATE  A    TURNOUT. 

3.  'Next  lay  off  the  proposed  line  of  the  near  rail  of  the  side 
track,  X  D.     Mark  the  point  G  on  that  line  where  the  range 
of  the  proper  gauge  side  of  frog  C  strikes  it.     Measure  C  G. 
Set  off  G  D,  equal  to  C  G,  along  the  side-track  line,  and  drive 
a  pin  at  D.     Stretch  the  cord  from  C  to  D.     Mark  the  middle 
point  of  it  at  K,  and  drive  a  pin  at  N,  half-way  between  K  and 
G.     Stretch  the  cord  from  C  to  N,  and  from  N  to  D.     Stick 
pins  outside  the  middle  points,  M  and  O,  of  those  stretches  at 
a  distance  from  those  points,  M  and  O,  equal  to  one-quarter  of 
KN. 

4.  These  three  sets  of  pins  will  fix  the  line  of  one  rail  of  the 
turnout.     The  corresponding  rail  of  a  double  turnout  can  be 
laid  off  from  them,  if  required,  by  symmetrical  measurements. 

5.  In  the  case  of  a  single  turnout,  stretch  the  cord  from  the 
toe  of  switch,  as  above,  to  the  point  of  frog,  located  by  the 
foregoing  tables ;  catch  it  at  the  middle,  and  pull  it  outwards 
to  a  point  in  range  with   the  line   of  the  switch-rail  in  one 
direction,  and  the  gauge  side  of  frog  in  the  other  direction. 
Half-way  between  that  point  and  the  middle  of  the  cord,  when 
straight,  stick  a  pin.     Measure   that  half-way  distance,   and 
divide    it  by  4;    call  the    quotient  the   "quarter-distance." 
Stretch  the  cord  from  the  pin  just  set  to  the  toe  of  switch  in 
one  direction,  and  to  the  point  of  frog  in  the  other.     Outside 
the  middle  points  of  these  short  stretches,  lay  off  the  "  quarter- 
distance,"  as  above  found,  and  stick  two  other  pins.     These 
three  pins  will  sufficiently  mark  the  line  of  the  outer  rail  of 
the  turnout. 

6.  The  same  methods  will  apply  in  practice  to  turnouts  from 
curves.     In  the  latter  case,  the  distance  C  G  is  to  be  calculated 
as  follows :  — 

Multiply  the  distance  Y  D,  between  the  nearest  rails  of  the 
parallel  tracks,  by  the  number  of  the  frog,  taken  from  the  fore- 
going table.  Thus,  on  the  full  gauge,  with  a  space  between 
tracks  of  7  feet  and  a  No.  0  frog,  the  distance  C  G  would  be  7 
X  6  ==  42  feet.  Lay  off  C  G,  in  range  of  the  gauge  side  of  the 
frog,  and  stick  a  pin  at  G.  Measure  out  G  D,  equal  to  C  G, 
and  set  another  pin  at  D,  making  D  Y  the  proper  distance  be- 
tween tracks.  Then  stretch  the  cord  from  D  to  C,  and  pro- 
ceed to  stake  off  the  curve  C  N  D,  as  above  directed. 


CROSSINGS   ON  STRAIGHT  LINES. 


139 


XLVIII. 

CROSSINGS   ON  STRAIGHT  LINES. 

1.  Having  located  frogs  B  and  C  by  the  preceding  methods, 
stretch  the  cord  any  convenient  distance,  C  D,  in  the  range  of 


the  outer  gauge  side  of  the  frog  C.  Set  off  E  F  parallel  to 
C  D,  and  distant  the  gauge-width  from  it.  The  intersection  of 
said  parallel  at  F  with  the  near  rail  of  the  side  track  marks 
the  spot  for  point  of  side-track  frog ;  the  curve  F  G,  thence  to 
toe  of  switch,  corresponds  to  AC  on  the  main  track,  and  may 
be  staked  out  in  like  manner. 


XLIX. 

CROSSINGS  ON  CURVES. 

1.  Having  located  frogs  B  and  C  by  the  preceding  methods, 
set  off  the  width  of  gauge,  CD,  from  point  of  frog  C,  and 
square  to  its  outer  gauge  side.     Stick  a  pin  at  D. 

2.  Next  calculate  the  distance  D  E  to  the  point  of  side-track 
frog  as  follows:    Subtract  the  gauge  of  track  from  the   dis- 


140 


CROSSINGS   ON  CURVES. 


tance,  HI,  between  the  gauge  sides  of  the  nearest  rails  of  the 
main  and  side  tracks;  multiply  the  remainder  by  the  number 


of  the  frog,  taken  from  foregoing  tables.     The  product  will  be 
the  distance  from  D  to  the  point  of  side-track  frog  at  E. 


ELEVATION  OF  THE  OUTER  RAIL   ON  CURVES.        141 

3.  Suppose,  for  example,  the  gauge  sides  of  the  nearest  rails 
of  the  main  and  side  tracks  are  6  feet  6  inches  asunder;  gauge 
of  track,  4  feet  8£  inches;  frog,  a  No.  9.  Reducing  inches  to 
decimals,  we  have  then  the  distance  between  tracks  6.5  feet, 
less  the  gauge,  4.7  feet,  =  1.8  feet;  and  1.8  multiplied  by  9, 
the  number  of  the  frog,  gives  16.2  feet  for  the  distance  D  E. 
The  proper  spring  will  be  given  to  rail  DE  on  the  ground;  and 
curve  E  G,  from  frog  to  toe  of  side-track  switch,  will  be  staked 
off  as  directed  in  the  section  on  turnouts. 


L. 

ELEVATION  OF  THE  OUTER  RAIL  ON  CURVES. 

1.  Great  precision  in  this  adjustment  is  unattainable,  owing 
to  differences  in  the  speed  of  trains  and  to  the  cost  of  track- 
maintenance,  if  it  were  attempted.     The  annexed  table  will  be 
found  convenient  in  practice.     It  has  been  calculated  by  the 
following  simple  rule:  — 

2.  Divide  the  speed  in  miles  per  hour  by  10;  multiply  the 
square  of  the  quotient  by  the  degree  of  curve.     The  product  is 
the  elevation  in  sixteenths  of  an  inch  for  full  gauge  of  4  feet 
8|  inches.     Take  two-thirds  of  it  for  3-feet  narrow  gauge. 

3.  Molesworth  gives  the  following  formula  for  determining 
the  elevation  of  the  outer  rail  with  any  gauge:  — 

V  =  greatest  velocity  of  trains  in  feet  per  second. 
G  =  gauge  of  railway  in  feet. 

C  =  length  of  chord  whose  middle  ordinate  will  give  the 
required  elevation. 

Then  C  =  \  V 

A  modification  of 
this  formula  gives  the 
following  approximate 
rules :  — 

To  fix  the  elevation 

of  the  outer  rail  on  the  standard  gauge  of  4  feet  8|  inches, 
multiply  the  speed  of  trains  in  miles  per  hour  by  5,  and  divide 


142         ELEVATION  OF  THE  OUTER  RAIL    ON  CURVES. 

the  product  by  3.  This  will  give  the  length  of  tape,  C,  to 
stretch  on  the  gauge  side  of  the  outer  rail ;  and  the  distance,  e, 
from  the  middle  of  the  tape  to  the  gauge  side  of  the  rail,  will 
be  the  proper  elevation. 

For  gauge  of  one  metre,  —  3.28  feet,  make  C  equal  to  one 
and  one-third  times  the  speed  of  trains  in  miles  per  hour. 

For  3-feet  gauge,  make  C  equal  to  one  and  one-fourth  times 
the  speed  of  trains  in  miles  per  hour. 


TABLE  OF  ELEVATIONS  FOR  OUTER  RAIL  ON  CURVES. 


SPEED   IN   MILES   PER  HOUR. 

10                             20                               30 

40 

H 

II                                  II 

•i 

fl 
b 

ELEVATION   OF  OUTER   RAIL  IN    INCHES  AND   FRACTIONS. 

: 

O 

o 

ll 

o 

|d 

1 

£d 

1 

&e 

| 

SO  C 

«T 
I 

o 

H 

W 
R 

J  9 

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CS-IC* 

3 

Jsr 

li 

H 

H 
Qj 

W 

5 

fi 

fr* 

fcw 

£"* 

£eo 

£"* 

t^  w 

£** 

^  M 

a 

1° 

1-16 

1-4 

3-16 

9-16 

3-8 

i 

11-16 

1° 

2° 

1-8 

.. 

1-2 

5-16 

1  1-8 

3-4 

2 

1  5-16 

2° 

3° 

3-16 

1-8 

3-4 

1-2 

1  11-16 

11-8 

3 

2 

3° 

4° 

1-4 

1 

11-16 

21-4 

1  1-2 

4 

2  11-16 

4" 

5° 

5-16 

1  1-4 

13-16 

2  13-16 

1  7-8 

5 

35-16 

5° 

6C 

3-8 

1-4 

1  1-2 

1 

33-8 

21-4 

6 

4 

6° 

7° 

7-16 

1  3-4 

3-16 

3  15-16 

25-8 

7 

4  11-16 

7° 

8° 

1-2 

2 

5-16 

41-2 

3 

8 

5  5-16 

8° 

9° 

9-16 

3-8 

21-4 

1-2 

51-16 

33-8 

9 

6 

9° 

10° 

5-8 

.  . 

2  1-2 

11-16 

5  10-16 

33-4 

10 

611-16 

10° 

11° 

11-16 

23-4 

13-16 

(i  3-16 

4  1-8 

11 

7  5-16 

11° 

12° 

3-4 

1-2 

3 

2 

6  3-4 

4  1-2 

12 

8 

12° 

13° 

13-16 

31-4 

23-16 

75-16 

47-8 

13 

811-16 

13° 

14° 

7-8 

31-2 

2  5-16 

7  14-16 

5  1-4 

14 

95-16 

14° 

15° 

15-16 

5-S 

33-4 

21-2 

87-16 

5  5-8 

15 

10 

15° 

16° 

1 

4 

2  11-16 

9 

6 

16 

10  11-16 

16° 

TRAQKMEN*8   TABLE  OF  CURVES. 


143 


LI. 


DEGREE  OF  CURVE. 

OOCC^OC.^COtOMO^OC-.CgC.^WtOM 

||1| 

to*,      ik  to                  w  to  to  to 

MM*M**«««Bm«-«. 

100  Feet. 

MIDDLE  ORDINATES  IN  INCHES  AND  FRACTIONS,  THE 
LENGTH  OF  CHORD  BEING 

tO  4-  i'  tO        4-4-        tO        4-  tO        4-4-        tO         4-  tO 

50  Feet. 

WtOMMOCO<000-^05WOi4^*KWtOlOr- 

OO  1-0  OO  00  lO  GO  4-  tO  OO  4-  lO  »  4-  GO         4-  do         4-  GO 

g 

? 

to 

1 

5-    "332S---SSSE- 

^010>              g—                    0105                     003 

20  Feet. 

sss|.*.-*.sgS  —  —  3S- 

*- 

1 
LENGTH  OF  CHORD  IN  FEET  AND 
INCHES,  WITH  A  MIDDLE 
ORDINATE  EQUAL  TO  GAUGE  OP 
TRACK. 

SS2SSSJ 

P 

'      85 

1 

B-i-So»B»S^o»..B^So? 

DEGREE  OP  CURVE. 

SS°8-J«««!'S«!>S-.'?S°81».«!'!<««St 

144 


TRACKMEN'S   TABLE  OF  CURVES. 


EXPLANATION  OF  THE  FOREGOING  TABLE. 

Columns  1  and  10  give  the  degree  of  curve. 

The  use  of  column  2,  containing  the  deflection  distances, 
may  be  illustrated  thus:  Suppose  stakes  4,  5,  and  6  to  be  miss- 
ing from  a  3-degree  curve,  and  that  stakes  2  and  3  are  still 
standing  100  feet  apart.  To  replace  the  missing  stakes,  pro- 
ceed as  follows:  Measure  100  feet  from  3  to  A,  and  make  a 
mark  at  A  exactly  in  range  with  2  and  3.  Find,  in  column  2 
of  the  table,  the  deflection  distance  for  a  3-degree  curve,  which 
is  seen  to  be  5  feet  3  inches.  Hold  one  end  of  the  tape  at  A, 


and,  stretching  5  feet  3  inches  towards  4,  nearly  square  to  the 
range  A-3,  make  a  scratch  on  the  ground  three  or  four  feet 
long,  swinging  the  tape  around  A  as  a  centre.  Next  lay  off 
100  feet  from  stake  3  to  the  scratch;  where  the  end  of  that 
measurement  strikes  it,  is  the  place  for  stake  4.  By  measuring 
100  feet  out  to  B  on  the  range  3-4,  and  proceeding  in  like 
manner,  stake  5  may  be  set;  and  so  on. 

3.  If  the  centre  line  is  already  staked  for  track  at  points  100 
feet  asunder,  and  the  degree  of  curve  is  wanted,  range  out  the 
straight  line  between  stakes,  as  above,  to  A  or  B,  and  measure 
across  from  those  marks  to  the  neighboring  location-stake. 
Suppose  the  distance  B-5,  for  example,  to  be  8  feet  9  inches. 
Referring,  then,  to  column  2  of  the  table,  we  find  that  deflec- 
tion distance  to  indicate  a  5-degree  curve.  If  the  distance 


TRACKMEN'S   TABLE  OF  CURVES.  145 

proved  to  be  4  feet  4  inches,  we  should  soon  discover  that  that 
distance  was  about  half-way  between  3  feet  6  inches  and  5  feet 
3  inches,  the  nearest  numbers  in  the  table  corresponding 
respectively  to  a  2-degree  and  a  3-degree  curve,  and  showing 
the  located  line  to  be  a  2£-degree  curve. 

4.  Let  AC  B  in  the  figure,  which  is  drawn  very  much  out  of 
proportion  in  order  to  make  the  subject  clear,  represent  the 
centre  line  of  a  curve.  Suppose  G  H  to  be  a  chord  100  feet 
long,  and  G  C  or  C  H  to  be  a  chord  50  feet  long.  Then  column 
3  in  the  table  gives  the  distance,  C  D,  from  the  middle  of  the 
100-feet  chord  to  the  rail,  and  column  4  gives  the  distance, 
E  F,  from  the  middle  of  the  50-feet  chord  to  the  rail,  for  the 
different  degrees  of  curve.  By  the  aid  of  these  columns,  pins 
can  be  set  25  feet  apart  on  a  curve  where  the  location-stakes 
are  100  feet  apart.  Thus,  for  a  3-degree  curve,  C  D  is  8  inches, 


\ 


and  E  F  2  inches.  If  pins  were  wanted  at  the  half-way  marks, 
N,  their  distance  from  the  dotted  short  chords  would  be  one- 
quarter  of  E  F.  It  must  be  an  uncommon  case,  however,  that 
calls  for  stakes  closer  together  than  25  feet. 

5.  Columns  5,  (5,  and  7  give  the  spring  of  rails  of  different 
lengths  for  the  various  degrees  of  curve. 

6.  Columns  8  and  9  give  figures  for  finding  the  degree  of 
curve,  by  simple  measurement  of  a  straight  line  on  the  track, 
as  follows:  Suppose  A  C  B  and  KIL  to  represent  the  rails  of  a 
curving  track.     From  any  point  A,   on  the  outer  rail,  sight 
across  to  a  point  B,  on  the  same  rail,  along  a  line  just  touching 
the  inner  rail  at  I.     Measure  from  A  to  B,  and  seek  the  dis- 
tance in  column  8  or  0,  according  to  the  gauge  of  track.     If 
the  distance,  for  example,  measured  232  feet  on  the  full  gauge, 
then  the  curve  would  be  a  4-degree  curve;  if  249  feet,  then  it 
would  indicate  a  3^-degree  curve,  for  the  reason  that  the 


146  TKACKMSN*B   TABLE  OF  CURVES. 

measured  distance  falls  half-way  between  the  distances  corre- 
sponding to  a  3-degree  and  a  4-degree  curve  respectively. 

7.  The  rate  of  curve  can  be  found  also  very  nearly  by  moans 
of  column  3.     To  do  so,  stretch  a  straight  line,  100  feet  long, 
between  points  on  either  rail;  for,  though  they  seem  very  dif- 
ferent in  the  figure,  the  two  rails  of  a  track  have  practically 
the  same  curvature.     Measure  from  the  middle  of  the  line 
across  to  the  gauge  side  of  the  rail,  and  seek  the  measured 
distance  in  column  3:  opposite  to  it,  in  column  1,  will  be 
found  the  degree  of  curve. 

8.  If,  in  any  case,  the  exact  figures  sought  are  not  found  in 
the  table,  take  out  the  next  figure  less  and  the  next  greater. 
Subtract  one  from  the  other,  and  divide  the  remainder  by  4. 
Add  the  fourth  part  of  the  difference  between  them,  thus 
determined,  to  the  smaller  number,  and  compare  the  sum  with 
the  number  sought.     If  still  too  small,  add   another  fourth 
part ;  and  so  on  until  the  distance  or  the  degree  is  ascertained 
to  within  a  quarter  part. 

9.  Suppose,  for  instance,  a  deflection  distance  measures  5 
feet  7  inches.     The  nearest  tabular  numbers  are  5  feet  3  inches 
and  7  feet.^  Their  difference  is  21  inches,  one-fourth  of  which 
is  5£  inches.     Adding  5|  inches  to  the  smaller  number,  5  feet 
3  inches,  gives  5  feet  8|-  inches,  which  indicates  nearly  enough 
a  3j-degree  curve.     Again:  if  a  measurement  of  175  feet  is 
sought  in  column  9,  the  track  is  seen  at  once,  without  calcula- 
tion, to  be  a  44-degree  curve. 


TABLES. 


TABLES. 


TABLE  I. 

TIME  OF  MERIDIAN  PASSAGE  OF  NORTH  STAR  ABOVE  THE 
POLE,  FOR  THE  YEAR  1870. 


I 

)AY  OF  MONTH 

I  St. 

nth. 

2ISt. 

January             .         .         . 
February 

H.     M. 

6     26  P.M. 

4     23     * 

H.     M. 
5     46  P.M. 

3     44      ' 

H.     M. 

5     07   P.M. 
3    05     ' 

March 

2     33 

i     54      ' 

i     14     ' 

April 

o     31 

II      52  A.M. 

II      13  A.M. 

May 

10     33  A    i. 

9     54     ' 

9     15     ' 

June 
July                    .         .         . 

8     32 
6     34 

7     53     " 
5     55     ' 

7     i3     " 
5     16     ' 

August              .         .         . 
September 

4     33 
2     31 

3     54     , 

i     52 

?     i^     ^ 

October    .... 

o     33 

II      50   P.M. 

II       II     P.M. 

November 

10      27    P.M. 

9    48     ' 

9    °9     ' 

December 

8     29     " 

7     50     ' 

7.    ii     ' 

» 

i 

For  the  year  1880,  add  three  minutes  to  the  time;  for  1890,  add  7  minutes; 
for  1900,  add  11  minutes. 


149 


150    * 


TABLES. 


TABLE  II. 

TIME  OF  EXTREME  ELONGATIONS  OF  NORTH  STAR  FOR  THE 
YEAR  1870,  — LAT.  40°. 


DAY  OF  MONTH. 

First.                            Eleventh. 

Twenty-first 

TIME  OF  ELONGATIONS. 

Eastern. 

Western,  j   Eastern. 

Western. 

Eastern. 

Western. 

H.  M. 

H.   M. 

H.  M. 

H.  M. 

H.  M. 

H.  M. 

Jan. 

O.  32  P.M. 

0.24  A.M. 

II  .  52  A.M. 

11.40  P.M. 

1  1  .  1  3  A  .  M  . 

II  .OI  P.M. 

Feb. 

I0.2Q  A.M. 

I0.l8  P.M. 

9-50     " 

9-38     " 

9.11    " 

8-59     " 

Mar. 

8-39    " 

8.27     " 

8.00    " 

7.48     " 

7.20 

7.09     " 

April. 

6-37    " 

6.25    " 

5.53    " 

5.46     " 

5-19    ". 

5-07     " 

May. 

4-39 

4.27 

4.00 

3.48     " 

3.21    " 

3-°9     " 

June. 

2.38    " 

2.26  " 

1.58    " 

1.47     " 

1.19    " 

1.07     " 

July. 

o-39    ' 

0.28    " 

11.57  P-M- 

11-49  A.M. 

II.  18  P.M. 

II  .09  A.M. 

Aug. 

10.35  P.M. 

10.27  A.M. 

9-56    " 

9.48     "          9.16     " 

9.08      " 

Sept. 

8-33    " 

8.25     " 

7-54    " 

7.46     "           7.14     " 

7.06      " 

Oct. 

6-35    " 

6.27     « 

5.56    " 

5-48  "  !  5-17  " 

5-09      " 

Nov. 

4-33    " 

4.26     " 

3-54    " 

3.46        1  3.15   ' 

3-07      " 

Dec. 

2-35    " 

2.27    " 

1.56    " 

1.48  "     1.17  " 

1.09      " 

For  the  year  1880,  add  3  minutes  to  the  time  ;  for  1S90,  add  7  minutes  ;  for 
1900,  add  11  minutes.    Change  of  latitude  affects  the  tabular  time  very  slightly. 


TABLE  III. 


AZIMUTHS  OF  THE  NORTH  STAR,  AND  THEIR  NATURAL 
TANGENTS. 


tii 

YEAR. 

D 
H 
H 

1870. 

1880. 

1890.          1900. 

< 

Azi- 

Nat. 

Azi- 

Nat. 

Azi- 

Nat. 

Azi- 

Nat. 

muth. 

Tan. 

muth. 

Tan. 

muth. 

Tan. 

n  uth. 

Tan. 

/ 

3° 

•36 

.02793 

•32 

.02677 

.28 

.02560 

•25 

•02473 

32 

•38 

.02851 

•34 

•02735 

•30 

.02619 

•27 

.02531 

34 

.40 

.02910 

•36 

.02793 

•32 

.02677 

.29 

.02589 

36 

•43 

.02997 

•39 

.02881 

•35 

.02764 

•3i 

.02648 

38 

•45. 

•°3°55 

.41 

.02939 

•37 

.02822 

•33 

.02706 

40 

.48 

•93143 

•44 

.03026 

.40 

.02910 

•36 

.02793 

42 

•52 

•03259 

•47 

.03114 

•43 

.02997 

•39 

.02881 

.   44 

•55 

•03346 

•5i 

.03230 

.46 

.03084 

.42 

.02968 

46 

•59 

•03463 

•55 

•03346 

•5° 

.  03201 

.46 

.03084 

48 

.04 

.03609 

•59 

.03463 

•54 

•033!7 

•50 

.03201 

50 

.09  ! 

-03754 

.04 

.  03609 

•59 

•03463 

•54 

•°33*7 

TABLES. 


.    151 


TABLE   IV. 

ROODS   AND   PERCHES  IN  DECIMAL    PARTS   OF   AN  ACRE. 
One  Aicre  =  4  Roods  =  160  Perches  =  4,840  Square  Yards  =43,560  Square  Feet. 


ROODS. 

PERCHES,  j 

ROODS. 

0 

1 

3 

3 

0     1 

-  3 

3 

.000 

.250 

.500 

•75° 

21 

•  131 

•38l 

•  631 

.881 

.006 

.256 

.506 

.756 

22 

•137 

•387 

•637 

.887 

.012 

.262 

.512 

.762 

23 

.144 

•394 

•644 

•894 

.019 
.025 
.031 

.269 
.275 
.28l 

•519 

•525 
•531 

•769 

•775 
.781 

24 

3 

.150 
.156 
.162 

.400 
.406 
.412 

.650 
.656 
.662 

•9°? 
.906 
.912 

•°37 

.287 

•537 

.787 

27 

.169 

.419 

.669 

.919 

.044 

.294 

•544 

•794 

28 

•i75 

•425 

•675 

•925 

.050 

.300 

•55° 

.800 

29 

.  181 

.431 

.681 

.931 

.056 

.306 

•556 

.806 

3° 

.187 

•437 

.687 

•937 

.062 

.312 

•  562 

.812 

3i 

.194 

•444 

.694 

•944 

.069 

.319 

•569 

.819 

32 

.200 

•45° 

.700 

•95° 

•°75 

•325' 

•575 

.825 

33 

.206 

.456 

.706 

.956 

.081 

•331 

.581 

.831 

34 

.212 

.462 

.712 

.962 

.087 

•337 

•587 

•837 

35 

.219 

.469 

.719 

.969 

.094 

•344 

•594 

.844 

36 

.225 

•475 

•725 

•975 

.  oo 

•35° 

.600 

.850 

37 

.231 

.481 

•731 

.981 

.  06 

•356 

.606 

•  856 

38 

•237 

.487 

•737 

.987 

.  12 

.362 

.612 

.862 

39 

.244 

•494 

•744 

•994 

•  19 

•369 

.619 

.869 

40 

.250 

.500 

•750 

i  .000 

•  25 

•375 

.625 

.875 

TABLE   V. 

DECIMALS  OF  AX  ACRE  IN  ONE  CHAIN  LENGTH  OF  100  FEET 
AND   OF   VARIOUS   WIDTHS. 


!  Width  in 
Rods. 

Decimals  of  an 
Acre  per  100 
Feet. 

Acres  per 
Mile. 

Width  in 
Rods. 

Decimals  of  an 
Acre  per  100 
Feet. 

Acres  per 
Mile.      i 

•   % 

•018939 

i 

5% 

•208333 

ii 

i 

.037879 

2 

6 

.227273 

„       12 

1  1^ 

.056818 

3 

6% 

.246212 

2 

•075757 

4 

7 

•265151 

14 

3 

.094697 
.113636 

1 

8  2 

.284091 
.303030 

15 

16 

3» 

.-132576 

7 

B% 

.321970 

17 

4 

.151515 

8 

9 

.340909 

18 

4/^ 

.170454 

9 

9/^ 

.359848 

19 

5 

.189394 

10 

10 

.378788 

20 

152 


TABLES. 


TABLE   VI. 

ACRES,  ROODS,  AND  PERCHES  IX  SQUARE  FEET." 


Acres. 

Square  Feet. 

Roods. 

Square  Feet. 

Perches. 

Square  Feet. 

i 

4356<> 

i 

10890 

T7 

4628.25 

2 

87120 

2 

21780 

18 

4900.50 

3 

130680 

3 

32670 

^9 

5!72-75 

4 

174240 

4 

4356° 

20 

5445-oo 

5 

217800 

21 

5717.75 

6 

261360 
304920 

Perches. 

Square  Feet. 

22 
23 

5989-50 
6261.75 

8 

348480 

24 

6534.00 

9 

392040  • 

i 

272.25 

25 

6806.25 

10 

435600 

2 

544-5° 

26 

7078.50 

ii 

479160 

3 

816.75 

27 

7350-75 

12 

522720 

4 

1089.00 

28 

7623.00 

13 

566280 

1361.25 

29 

7895-25 

*4 

609840 

6 

^S.so 

30 

8167.50 

15 

653400 

7 

I905-75 

31 

8439-75 

16 

696960 

8 

2178.00 

32 

8712.00 

17 

740520 

9 

2450.25 

33 

8984.25 

18 

783080 

10 

2722.50 

34 

9256.50 

*9 

827640 

ii 

2994-75 

35 

9528.75 

20 

871200 

12 

3267.00 

36 

9801.00 

21 

916760 

13 

3539-25 

37 

10073.25 

22 

960320 

H 

3811.50 

38 

10345-50 

15 

4083.75 

39 

10617.75 

16 

4356.00 

40 

10890.00 

TABLE   VII. 

CIRCULAR  ARCS  TO  RADIUS  OF  1. 


DEGREES. 

'  MINUTES. 

SECONDS. 

i 

.01745329 

/ 

i 

.00029089 

// 

.00000485 

2 

.03490658 

2 

.00058178 

2 

.00000970 

3 

.05235988 

3 

.00087266 

3 

.00001454 

4 

.06981317 

4 

.00116355 

4 

.00001939 

5 

.08726646 

.00145444 

5 

.00002424 

6 

.10471975 

6 

•00174533 

6 

.00002909 

7 

.12217305 

7 

.00203622 

7 

.00003394 

8 
9 

.13962634 
.15707963 

8 
9 

.00232711 
.00261799 

8 
9 

.00003878 
.00004363 

TABLES. 


153 


TABLE  VIII. 

FEET  IN  DECIMALS  OF  A  MILE. 


Feet. 

Dec 

in 

als  of  a  Mile. 

I 

o 

0 

0 

0 

I 

8 

9 

3 

9 

4 

2 

o 

0 

0 

0 

3 

7 

8 

7 

9 

8 

3 

o 

0 

0 

0 

5 

6 

8 

i 

8 

2 

4 

o 

0 

0 

0 

7 

5 

7 

5 

7 

6 

5 

o 

0 

0 

0 

9 

4 

6 

9 

7 

0 

6 

o 

0 

0 

I 

i 

3 

6 

3 

6 

4 

7 

o 

0 

0 

I 

3 

2 

5 

7 

5 

8 

8 

0.0 

0 

I 

5 

I 

5 

i 

5 

2 

9 

o 

0 

0 

I 

7 

O 

4 

5 

4 

6 

TABLE   TX. 

INCHES  REDUCED  TO  DECIMAL  PARTS  OF  A  FOOT. 


In. 

O 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

ii 

In. 

0 

.OOOO 

•0833 

.1667 

.2500 

•3333 

.4167 

.5000 

•5833 

.6667 

.7500 

•8333 

.9167 

0 

re 

.0052.0855 

.1719 

•2552 

.3385 

.4219 

•5052 

•5885 

.6719 

•7552 

•8385 

.9219 

A 

| 

.0104.0938 

.1771 

.2604 

•3438 

.4271 

.5104 

•5938 

.6771 

.7604 

•8438 

.9271 

^ 

136 

.01561-0990 

.1823 

.2656 

•3490 

•4323 

•5156 

•599° 

.6823 

.7656 

.8490 

•9323 

T3g 

i 

.0208 

.1042 

.1875 

.2708 

•3542 

•4375 

.5208 

.6042 

.6875 

.7708 

.8542 

•9375 

\ 

156 

.026oj.  1094 

.1927 

.2760 

•3594 

.4427 

.5260 

.6094 

.6927 

.7760 

•8594 

•9427 

T56 

| 

.0313 

.II46j.I979 

.2813 

.3646 

•4479 

•5313 

.6146 

.6979 

•7813 

.8646 

•9479 

-| 

7 
16 

•0365 

.1198  .2031 

.2865 

.3698 

•4531 

•5365 

.6198 

.7865 

.8698 

•9531 

A 

i 

.0417 

.1250 

.2083 

.2917 

•3750 

.45831.5417 

.6250 

.7083 

.7917 

.8750 

•9583 

T96 

.0469 

.1302 

•2135 

.2969 

.3802 

•4635!  -5469 

.6302 

•7^35 

.7969 

.8802 

•9635 

IT 

1 

.0521 

•1354 

.2188 

.3021 

•3854 

.4688 

•5521 

•6354 

.7188 

.8021 

.8854 

.9688 

I 

H 

•0573 

.1406 

.2240 

•3°73 

.3906 

.4740 

•5573 

.6406 

.7240 

•8073 

8906 

974° 

u 

I 

.0625 

.1458 

.2292 

•3125 

.3958 

.4792 

•5625 

.6458 

.7292 

.8125 

8958 

•9792 

1 

1  3 
1  15 

.0677 

.1510 

•2344 

•3177 

.4010 

.4844 

•5677 

.6510 

•7344 

.8177 

.9010 

9844 

\l 

7 

8 

.0729 

.1563  .2396^3229 

.4063 

.4846 

•5729 

•6563 

.7396  .8229 

.9063 

9896 

1 

8 

ii 

.0781 

.1615 

.2448 

.3281 

•4»3 

•4948 

•578i 

.6615 

.7448 

.8281 

.9115 

9948 

ii 

TABLE    X. 

RADII   AND   THEIR  LOGARITHMS,    MIDDLE   ORDI- 
NATES,   AND   DEFLECTION  DISTANCES. 


156 


RADII  AND    THEIR  LOGARITHMS. 


:  Degree 
of 
!  Curve. 

Radius. 

Logarithm 
of 
Radius. 

Arithmetical 
Comple- 
ment. 

Middle 
Ordinate, 
Chord 
100  Feet. 

Deflec- 
tion Dis- 
tance. 

Tangen- 
tial Dis- 
tance. 

3  5' 

68754.9 

4-837304 

5.162696 

.018 

•I45 

•°73     ! 

10 

34377-5 

4.536274 

5.463726 

.036 

.291 

•MS     i 

15 

22918.3 

4.360182 

5.639818 

•°55 

•436 

.218     | 

20 

17188.8 

4.235246 

5-764754 

•°73 

•  582 

.291 

25 

I375I-o 

4-138335 

5.861665 

.091 

.727 

'•364 

3° 

11459.2 

4.059154 

5.940846 

.109 

•873 

•436 

35 

9822.2 

3.992209 

6.007791 

.127 

i  .02 

•5°9 

46 

8594.4 

3-9342I5 

6.065785 

•145 

1.  16 

•  582 

45 

7639-5 

3.883066 

6.116934 

.164 

I-3T 

•654 

50 

6875-5 

3-837304 

6.162696 

.182 

1.45 

.727 

55 

6250.5 

3.795914 

6  .  204086 

.200 

i.  60 

.800 

1       0 

5729.6 

3.758128 

6.241872 

.218 

T-75 

•873 

5 

5288.9 

3-723365 

6.276635 

•236 

1.89 

•945 

10 

4911.1 

3.691179 

6.308821 

•255 

2.04 

1.02 

15 

4583-7 

3.661216 

6.338784 

•273 

2.18 

1.09 

20 

3-633I95 

6.366805 

.291 

2-33 

1.16 

25 

4044.5 

3.606864 

6.393136 

•309 

2.47 

1.24 

3° 

3819.8 

3.582041 

6.417959 

•327 

2.62 

1.31 

35 

3618.8 

3-558565 

6-44M35 

•345 

2.76 

1.38 

40 

3437-9 

3-536293 

6.463707 

•364 

2.91 

45 

6.484894 

.382 

3-°5 

I-53 

50 

3125.4 

3.494906 

6.505094 

.400 

3-20 

i.  60 

55 

2989.5 

3-475599 

6.524401 

.418 

3-34 

1.67 

2     o 

2864.9 

3-457"4 

6.542886 

-436 

3-49 

1.74 

5 

2750-3           3-43938o 

6.560620 

•455 

3-64 

1.82 

10 

2644.6 

3.422359 

6.577641 

•473 

3-78 

1.89 

15 

2546.6 

3.405961 

6.594039 

.491 

3-93 

1.96 

20 

2455-7 

3-390I75 

6.609825 

•509 

4.07 

2.04 

25 

2371.0 

3-374932 

6.625068 

•527 

4.22 

2.  II 

3° 

2292.0 

3.360215 

6-639785 

•545 

4-36 

2.18 

35 

2218.  i 

3.345982 

6.654018 

•564 

4-51 

2.25 

40 

2148.8 

3-332196 

6.667804 

-582 

4-65 

2-33 

45 

2083.7 

3-318835 

6.681165 

.600 

4.80 

2.40 

50 

2022.4 

3.305867 

6.694133 

.618 

4-94 

2-47 

55 

1964.6 

3.293274 

6.706726 

-636 

5-09 

2-54 

3     o 

1910.1 

3.281056 

6.718944 

•655 

5-23 

2.62 

5 

1858.5 

3.269163 

6.730837 

•673 

5-38 

2.69 

10 

i  809  .  6 

3.257584 

6.742416 

.691 

5-53 

2.76 

15 

1763.2 

3.246301 

6-753699 

.709 

5-67     - 

2.84 

20 

1719.1 

3.235301 

6.764699 

.727 

5-82 

2.91 

25 

1677.2 

3.224585 

6.775415 

•745 

5.96 

2.98 

3° 

1637.3 

3.214129 

6.785871 

•764 

6.11 

3-05 

35 

1599.2 

3.203902 

6.796098 

.782 

6.25 

40 

1562.9 

6.806069 

.800 

6.40 

3-20 

45 

1528.2 

3.  184180 

6.815819 

.818 

6-54 

3-27 

50 

1495.0 

3.174641 

6.825359 

•  836 

6.69 

3-34 

55 

1463.2 

3.165303 

6.834607 

•855 

6.83 

3-42 

4    o 

1432.7 

3-i56i55 

6.843845 

.873 

6.98 

3-49 

5 

3.147212 

6.852788 

.891 

7.12 

3-56 

10 

1375-4 

3-138429 

6.861571 

.909 

7-27 

3-63 

15 

1348.4 

3.129819 

6.870181 

•927 

•7.42 

3-71 

20 

1322.5 

3-121395 

6.878605 

•945 

7-56 

3.78 

25 

1297.6 

6.886859 

.964 

7.71 

3-85 

30 

1273.6 

3-105033 

6.894967 

.982 

7-85 

3-93 

35 

1250.4 

3.097048 

6.902952 

I.  00 

8.00 

4.00 

40 

1228.1 

3-089233 

6.910767 

i.  02 

8.14 

4.07 

RADII  AXD    THEIR  LOGARITHMS. 


157 


Degree 
of 
Curve. 

Radius. 

Logarithm 
of 
Radius. 

Arithmetical 
Comple- 
ment. 

Middle 
Ordinate, 
Chord 
100  Feet. 

Deflec- 
tion Dis- 
tance. 

Tangen- 
tial Dis- 
tance.    | 

i 

o     / 
4  45 

206.6 

3-081563 

6.918437 

1.04 

8.29 

4.14 

5° 

185.8 

3.074011 

6.925989. 

05 

8-43 

4.22 

55 

l65-7 

3.066587 

6.933413 

07 

8.58 

4.29 

5     o 

146.3 

3.059299 

6.940701 

09 

8.72 

•36 

5 

i27-5 

3.052117 

6.947883 

.11 

8.87 

•43 

10 

109.3 

3.045050 

6.954950 

•J3 

9.01 

•5i 

15 

091.7 

3.038103 

6.961897 

15 

9.16 

-58 

20 

074.7 

3.031287 

6.968713 

.16 

9-3° 

-65 

25 

058.2 

3.024568 

6-975432 

.18 

9-45 

.72 

30 

042.1 

3.017910 

6.982090 

.20 

9.60 

.80 

35 

026.6 

5.011401 

6.988599 

.22 

9-74 

•87 

40 

on.  5 

3.004967 

6.995033 

.24 

9.89 

•94 

45 

996.9 

2.998652 

7.001348 

•25 

IO.O 

5.02 

5° 

982.6 

2.992377 

7.007623 

•27 

10.2 

5-°9 

55 

968.8 

2.986234 

7-013766 

.29 

10.3 

5-i6 

G     o 

955-4 

2.980185 

7.019815 

•31 

10-5 

5-23 

5 

942-3 

2.974189 

7.025811 

•33 

10.6 

5-3i 

10 

929.6 

2.968296 

7.031704 

•35 

10.8 

5.38 

15 

917.2 

2.962464 

7.037536 

•3£ 

10.9 

5-45 

20 

§5-  r 

2.956697 

7-043303 

'    -38 

1  1  .0 

S-S2 

25 

3-4 

2.951046 

7.048954 

.40 

II.  2 

5-6o 

3° 

2.0 

2.945469 

7-05453I 

.42 

"•3 

5-67 

35 

870.8 

2.939918 

7.060082 

•44 

n-5 

5-74 

40 

859.9 

2.934448 

7-065552 

•45 

n  .6 

5-8i 

45 

849-3 

2.929061 

7.070939 

•47 

ii.  8 

5-89 

5° 

839.0 

2.923762 

7.076238 

•49 

11.9 

5.-  96 

55 

828.9 

2.918502 

7.082498 

•51 

12.  I 

6.03 

7    o 

819.0 

2.913284 

7.086716 

•53 

12.2 

6.  io 

5 

809.4 

2.908163 

7.091837 

•55 

12.3 

6.18 

10 

800.0 

2.903090 

7.096910 

•56 

12.5 

6.25 

1  5 

790.8 

2.898067 

7-IOI933 

-58 

12.6 

6.32 

20 

781.8 

2.893096 

7.106904 

.60 

12.8 

6-39 

25 

773-1 

2.888236 

7.111764 

.62 

12.9 

6-47 

30 

764-5 

2-883377 

7.116623 

.64 

13-1 

6-54 

35 

756-I 

2.878579 

7.121421 

.65 

13.2 

6.61 

40 

747-9 

2.873844 

7.126156 

.67 

13-4 

6.68 

45 

739-9 

2.869173 

7.130827 

.69 

13-5 

6.76 

5« 

732.0 

2.864511 

7-I35489 

•7t 

!3-7 

6.83 

55 

724-3 

2.859918 

7  .  i  40082 

•73 

13-8 

6.90 

8     o 

716.8 

2-855398 

7.144602 

•75 

14.0 

6.98 

5 

709.4 

2.850891 

7.149109 

•76 

14.1 

7-°5 

io 

702.2 

2.846461 

7-^53539 

.78 

14.2 

7.12 

15 

695.1 

2.842047 

7-I57953 

.80 

14.4 

7.19 

20 

688.2 

2-837715 

7:162285 

.82 

14-5 

7.27 

25 

681.3 

2-833338 

7.166662 

.84 

14.7 

7-34 

3° 

674.7 

2.82QI  I  t 

7.170889 

.85 

14-8 

7.41 

35 

668.1 

2.82484I 

7-I79*59 

.87 

15.0 

7.48 

40 

661.7 

2.820661 

7-I79339 

.89 

15-1 

7-56     j 

45 

655-  '4 

2.816506 

7.183494 

.91 

15-3 

7-63 

5° 

649-3 

2.8l2445 

7-  I87555 

•93 

i5-4 

7.70 

55 

643.2 

2.808346 

7-181654 

•95 

iS-5 

7-77      1 

9     o 

637-3 

2.804344 

7.195656 

.96 

«5-7 

7-85 

5 

631.4 

2.800305 

7.199695 

.98 

15.8 

7-92 

10 

625.7 

2.796366 

7.203634 

.00 

16.0 

7-99 

J5 

620.  i 

2.792462 

7-2U7538 

.02 

16.1 

8.06 

RADII  AND    THEIR  LOGARITHMS. 


Degree 
of 
Curve. 

Radius. 

Logarithm 
of 
Radius. 

Arithmetical 
Comple- 
ment. 

Middle 
Ordinate, 
Chord 
loo  Feet. 

Deflec- 
tion Dis- 
tance. 

Tangen- 
tial 'Dis- 
tance. 

?j2'0 

614.6 

2.788593 

7.211407 

2.04 

16.3 

8.14 

25 

609.1 

2.784689 

7-2T5311 

2.06 

16.4 

8.21 

30 

603.8 

2.780893 

7.219107 

2.07 

16.6 

8.28 

35 

598.6 

2.777137 

7.222863 

2.09 

16.7 

8-35 

40 

593-4 

2.773348 

7.226652 

2.  II 

16.8 

8.43 

45 

588.4 

2-769673 

7.230327 

2.13 

17.0 

8.50 

5° 

583-4 

2.765966 

7-234I34 

2.15 

17.1 

8-57 

55 

578-5 

2.762303 

7-237697 

•2.l6 

17-3 

8.64 

1O     o 

573-7 

2.758685 

7-24I3i5 

2.l8 

17.4 

8.72 

10 

564-3 

2.751510 

7.248490 

2.22 

17.7 

8.86 

20 

555-2 

2-744449 

7-25555I 

2.26 

18.0 

9.00 

30 

546-4 

2-7375" 

7.262489 

2.29 

18.3 

9.15 

40 

537-9 

2.730702 

7.269298 

2-33 

18.6 

9-30 

5° 

529-7 

2.724030 

7.275970 

2.36 

18.9 

9-44 

11       0 

521-7 

2.717421 

7.282579 

2.40 

19.2 

9-58 

IO 

5x3-9 

2.710879 

7.289121 

2.44 

19-5 

9  -73 

20 

506.4 

2.704494 

7-295506 

2-47 

19.7 

9.87 

3° 

499.1 

2.698188 

7.301812 

2-51 

20.  o 

IO.O 

40 

492.0 

2.691965 

7-308035 

2-55 

20.3 

10.2 

5° 

485.1 

2.685831 

7.314169 

2.58 

20.6 

10-3 

12     o 

478-3 

2.679700 

7.320300 

2.62 

20.9 

10.4 

IO 

471.8 

2-673758 

7.326242 

2.66 

21  .2 

10.6 

20 

465-5 

2.667920 

7.332080 

2.69 

21-5 

10.7 

3° 

459-3 

2  .  662006 

7-337904 

2-73 

21.8 

10.9 

40 

453-3 

2.656386 

7-3436I4 

2.77 

22.1 

II.  0 

5° 

447-4 

2.650696 

7-349304 

2.80 

22.4 

II  .2 

13     o 

441.7 

2.645127 

7-354873 

2.84 

22.6 

"•3 

10 

436.1 

2.639586 

7.360414 

2.88 

22-9 

20 

43°  -7 

2-634175 

7.365825 

2.91 

23.2 

ii.  6 

30 

425-4 

2.628797 

7.371203 

2-95 

23-5 

11.7 

40 

420.2 

2.623456 

7.376544 

2.98 

23-8 

11.9 

50 

415.2 

2.618257 

7-38I743 

3-02 

24.1 

12.  0 

14     o 

410.3 

2.6I3I02 

7.386898 

3.06 

24-4 

12.2 

IO 

4°5-5 

2.607991 

7.392009 

3.09 

24.7 

I2.3 

20 

400.8 

2.602928 

7-397072 

3-13 

25.0 

12-5 

3° 

396.2 

2.597914 

7  .  402086 

3-T7 

25.2 

12.6 

40 

39r-7 

2092954 

7.407046 

3-20 

25-5 

12.8 

50 

387-3 

2.588047 

7-4"953 

3-24 

25-8 

12.9 

15     o 

383-1 

2-583312 

7.416688 

3-28 

26.1 

13.0 

10 

378-9 

2-578525 

7-42I475 

3-31 

26.4 

I3.2 

20 

374-8 

2.573800 

7.426200 

3-35 

26.7 

13-3 

30 

370.8 

2.569140 

7.430860 

3-39 

27.0 

13-5 

40 

366.9 

2.564548 

7-435452 

3-42 

27-3 

I3.6 

50 

363-0 

2.559907 

7.440093 

3-46 

27-5 

13-8 

16     o 

359-3 

2-555457 

7-444543 

3-50 

27.8 

13-9 

IO 

355-6 

2.550962 

7.449038 

3-53 

28.1 

I4.I 

20 

352-o 

2-546543 

7-453457 

3-57 

28.4 

14.2 

30 

348.4 

2  .  542078 

7.457922 

3.61 

28.7 

14-3 

40 

345-° 

sJTJ 

2.537819 

7.462181 

3-64 

29.0 

M-5 

50 

34'-  6 

2-5335'8 

7.466482 

3-68 

29-3 

14.6 

|   17     o 

338.3 

2.529302 

7.470698 

3-72 

29.6 

14-8 

10       i          335-0 

1 

2.525045 

7-474955 

3.75             ,.               14.9 

RADII  ANT)    7777-77?   LOGARITHMS. 


159 


!    Degree 
of 
1  ;    Curve. 

Radius. 

Logarithm 
of 
Radius. 

Arithmetical 
Comple- 
ment. 

Middle 
Ordinate, 
Chord 
loo  Feet. 

Deflec- 
tion Dis- 
tance. 

Tangen- 
tial Dis- 
tance. 

o      / 

17     20 

33'.8 

.520876 

7.479124 

3-79 

30.1 

J5-1 

3° 

328.7 

.516800 

7.483200 

3.82 

3°-4 

15.2 

40 

325-6 

.512684 

7.487316 

3-86 

3°-7 

J5-4 

5° 

322.6 

.508664 

J'WSS^ 

'3-9° 

31.0 

J5-5 

18     o 

319.6 

.504607 

7-495393 

3-93 

3T-3 

15-6 

10 

316.7 

.500648 

7-499352 

3-97 

31-6 

15-8 

20 

3*3-9 

.496791 

7.503209 

4.01 

3i-9 

15-9 

30 

311.1 

.492900 

7.507100 

4.04 

32.1 

16.1 

40 

308.3 

.488974 

7.511026 

4-08 

32-4 

16.2 

50 

305-6 

•485153 

7-5I4847 

4.12 

32-7 

16.4 

19     o 

302.9 

.481299 

7.518701 

4-I5 

33-° 

16.5 

10 

300.3 

•477555 

7.522445 

4.19 

33-3 

16.6 

20 

297.8 

•473925 

7.526075 

4-23 

33-6 

16.8 

3° 

295.2 

.470116 

7.529884* 

4.26 

33-9 

16.9 

40 

292.8 

.466571 

7-533429 

4-30 

34-2 

17.1 

50 

290.3 

.462847 

7-537I53 

4-34 

34-4 

17.2 

3O     o 

287.9 

2.459242 

7-540758 

4-37 

34-7 

17.4 

TABLE   XI. 

SQUARES,   CUBES,   ETC.,  OF  NUMBERS  • 
FROM   1    TO   1042. 


TABLE 


SQUARES,  CUBES,  SQUARE  AND  CUBE  ROOTS  OF  NUMBERS 


No. 

Squares.       Cubes. 

Sqimre  Roots. 

Cube  Roots. 

Reciprocals. 

1 

1 

1 

1-0000000 

1-0000000 

•100000000 

2 

4 

8 

1-4142136 

1-2599-210 

•500000000 

3 

9 

27 

1-7320508 

1-4422496 

•333333333 

4 

10 

64 

2-0000000 

1-5874011 

•250000000 

5 

25 

125 

2-2300f>80 

1-7099759 

•200000000 

6 

36 

216 

2-44948.-I7 

1  '8  171206 

•1666061567 

7 

49 

343 

2-0457513 

1-9129312 

•142857143 

8 

64 

512- 

2-8284271 

2-0000000 

•125000000 

B 

81 

729 

3-0000000 

2-0800837 

•111111111 

10 

100 

1000 

3-1622777 

2-1544347 

•100000000 

II 

Kl 

1331 

3-3100248 

2-2239801 

•090909091 

It 

144 

1728 

3-4041010 

2-2894286 

•083333333 

13 

169 

2197 

3-005.").-)  13 

2-3513347 

•070923077 

14 

190 

2744 

3-7410574 

2-4101422 

•071428571 

15 

225 

3375 

3-8729833 

2-4002121 

•006006607 

10 

256 

4090 

4-0000000 

2-5198421 

•002500000 

17 

289 

4913 

4-1231056 

2-5712816 

•0588-235-29 

18 

3-24 

5832 

4"24-2(>407 

2-62J7414 

•05555555(5 

19 

3til 

6859 

4-3588989 

2-0084016 

•052031579 

20 

400 

8000 

4-4721300 

2-7144177 

•050000000 

2! 

441 

9261 

4-5825757 

2-7589243 

•047(519048 

22 

484 

10048 

4-0904158 

2-8020393 

•045454545 

23 

529 

12167 

4-7958315 

2-8438070 

•043478261 

24 

570 

13824 

4-8989795 

2-8844991 

•041000007 

25 

625 

15625 

5-0000000 

2-9240177 

•040000000 

20 

670 

17576 

5-0990195 

2-9024900 

•038401538 

27 

729 

19683 

5-190152-1 

3-0000000 

•037037037 

28 

784 

21952 

5-2915020 

3-0305889 

•035714286 

29 

841 

24389 

5-3851648 

3-07-23108 

•034482759 

30 

900 

27000 

5-4772250 

3-107-23-25 

•033333333 

31 

901 

29791 

5-5077G44 

3-1413800 

•032-258005 

32 

1024 

32768 

5-6568542 

3-1748021 

•031250000 

33 

1089 

35937 

5-744.10-20 

3-2075343 

•030303030 

34 

1150 

39304 

5-8309519 

3-2390118 

•029411705 

35 

1225 

42875 

5-9100798 

3-2710003 

•028571429 

3G 

1296 

46656 

6-0000000 

3-3019-272 

•027777778 

37 

1309 

50053 

0-08-270-25 

3-33-2-2-218 

•0270270-27 

38 

1444 

54872 

6-1044140 

3-3019754 

•026315789 

39 

1521 

59319 

6-2449980 

3-3912114 

•025041026 

40 

1000 

64000 

6-3245553 

3-4199519 

•025000000 

41 

1081 

68921 

6-4031-242 

3-4482172 

•024390244 

42 

1704 

74088 

6-480741)7 

3-4700-260 

•0238095-24 

43 

1849 

7W507 

6'5.")74385 

3-5033981 

•023255814 

44 

1930 

85184 

6-0332490 

3-5303483 

•02-2727273 

45 

2025 

91125 

6-7082039 

3-550893:1 

•0222-2-2-2-22 

46 

2116 

97336 

6-7823300 

3-5830479 

•0-21739130 

47 

2209 

103823 

6-8550540 

3-0088201 

•021270000 

48 

2304 

110592 

6-9282032 

3-0342411 

•0-20833333 

49 

2401 

117049 

7-0001)1100 

3-0593057 

•020408103 

50 

2500 

125000 

7-0710678 

3-0840314 

•020000000 

162 


SQUARES,  CUBES,  ETC..  OF  NUMBERS. 


163 


Squares, 


Square  Rn 


Reciprocals. 


2001 

132051 

7-1414284 

•  3-7084298 

•019607843 

'  2:04 

140008 

7-211  1026 

3-7325111 

•019230709 

8806 

148877 

7-2801099 

3-7562858 

•018867925 

2»ffi 

157404 

7-3484092 

3-7797631 

•018518519 

3025 

100375 

7-4101985 

3-80-29525 

•OI818181H 

3130 

175010 

74833148 

3-8258624 

•017857143 

3249 

185193 

7-5498344 

3-8485011 

•01754:5860 

3364 

195112 

7-0157731 

3-H708766 

•017241379 

3481 

2(15379 

7-08  11  457 

3-8D29905 

•016949153 

3680 

210000 

7-7459667 

3-9148670 

•01(>06f;667 

3721 

2-20981 

7-8102497 

3-93J4972 

•016393443 

3844 

238328 

7-8740079 

3-9578915 

•01612903^ 

3909 

250047 

7-9372539 

3-9790571 

•015873016 

4096 

202144 

8-0000000 

4-0000000 

•0  156251  MM) 

4395 

274025 

8-0622577 

4-0207256 

•01  538401  5 

4356 

28749G 

8-1240384 

4-0412401 

•015151515 

4489 

3007(»3 

8-1853528 

4-0615480 

•011925373 

4624 

314432 

8-2402113 

4-0816551 

•014705832 

47G1 

328509 

8-3066239 

4-1015661 

•014492754 

4900 

343000 

8-3060003 

4-1212853 

•0142K>714 

5041 

357911 

8-4201498 

4-1408178 

•014084517 

5184 

373248 

8-4852814 

4-1001076 

•013888889 

533) 

389017 

8-5440037 

4-1793390 

•013698630 

5476 

405224 

86023253 

.  4-1983304 

•013513514 

50-25 

421875 

8-6002540 

4-2171033 

•013333333 

5776 

43897G 

8-7177979 

4-2358236 

•013157895 

stay 

450533 

8-7749044 

4-25432  10 

•012987013 

6084 

474552 

8-8317009 

4-2720586 

•012820513 

6241 

493039 

8-8881944 

4-2908404 

•0121)58228 

6400 

512000 

89442719 

4-3088695 

•012500000 

6561 

531441 

9-0000000 

4-3207487 

•012.-M5679 

6724 

551308 

9-0553851 

43444815 

012195122 

6889 

571787 

9-1104336 

4-3(320707 

•012048193 

7056 

592704 

9-1651514 

43795191 

•011904762 

7-225 

614125 

9-21!  (544  5 

4-3908296 

•011764706 

7396 

63CM50 

9-2736185 

4-4140049 

•011627907 

7569 

658503 

9-3273791 

4-4310476 

•011494253 

7,44 

081472 

9-3808315 

4-44791302 

•011303036 

7921 

704909 

9-4339811 

4-4647451 

•011235955 

8100 

729000 

94868330 

44814047 

Oil  111  111 

8281 

753571 

9-5393920 

4-4979114 

•01  09890  II 

84G4 

778(588 

95916030 

45143574 

•OI08695G5 

swy 

804357 

9-6436508 

4-5306549 

010752088 

8836 

830584 

9-6953597 

4-546H359 

•010038298 

9025 

857375 

9-7407943 

4-502!H)26 

•010520316 

92  1G 

88473G 

9-7979590 

4-5788570 

•010416607 

9409 

912073 

9-8488578 

45947009 

•010309273 

9604 

941192 

9-8994949 

4-6104303 

•010204082 

9801 

970299 

9-9498744 

4-0260050 

•010101010 

10000 

1000000 

100000000 

4-6415888 

01001)00110 

JU201 

1030301 

100498756 

4-6570095 

•00990091)0 

10404 

1061208 

100995049 

4-0723287 

•00!)8()3:»22 

10G09 

1092727 

10-1488916 

4-0875482 

•009708738 

1081G 

1124804 

10-1980390 

4-7020094 

oojGir>:j85 

11025 

1157625 

10-2469508 

47176940 

•009523810 

11236 

119101G 

10-2956301 

4-7326235 

•009433902 

11449 

1225043 

HKW40804 

4-7474594 

•009345794 

11(U)4 

1259712 

10-39-23048 

4-7622032 

•00;)25!)259 

11881 

1295029 

10-4403065 

4-77ti8502 

009174312 

12100 

1331000 

10-4880885 

4-7914199 

•009090909 

12321 

13G7031 

10-5356538 

4-8058995 

•009009009 

12544 

1404928 

10-5830052 

4-8202845 

•00-S928571 

104 


SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

113 

12769 

1442897 

106301458 

4-834.5881 

•00884!  (558 

114 

12U.96 

1481544 

10K770783 

4-8488076 

•QBH77IU9P 

115 

13225 

1520875 

10-7238053 

4-86-2944:2 

•OOH6950.V2 

JIG 

134.56 

1560896 

107703296 

4-87699!):) 

•008020690 

117 

13689 

1601613 

108166538 

4-89097:*-2 

•008547009 

118 

13<)-24 

1643032 

10-8627^05 

•9048681 

•OOS474576 

119 

14161 

1685159 

109087121 

9186847 

•008403361 

]20 

14100 

1728000 

10-9544512 

-.(M-24-24-2 

•008333333 

121 

14641 

1771561 

11  -0000000 

•9460874 

•008-264463 

122 

14834 

1815848 

11-0453610 

•9596757 

•008196721 

123 

15129 

1860867 

1]  0905365 

•9731898 

•008  130081 

J24 

15376 

1906624 

11  1355287 

•9866310 

•0080(545  16 

125 

15625 

1953125 

11-1803399 

5-0000000 

•00*000000 

126 

15876 

2000376 

1I-2-2497-2-2 

5-013297!) 

•007936508 

127 

16129 

2048383 

11-2694277 

5-0265-257 

•007874016 

128 

16384 

2097152 

11-3137085 

5-0396842 

•00781-2500 

129 

16641 

2146689 

11-3578167 

50527743 

•007751938 

130 

16900 

2197001) 

11-4017543 

5-0657970 

•007692308 

131 

17161 

2248091 

11-4455231 

5-0787531 

•007633588 

132 

17424 

2299968 

11-489I-2.-3 

5-0916434 

•007575758 

133 

17689 

2352637 

11-53-25626 

5-1044687 

•007518797 

134 

17956 

2406104 

11-5758369 

5  117-2-209 

•007462687 

135 

18225 

2460375 

11-6189500 

5-1299-278 

•007407407 

136 

18496 

2515456 

H-6619038 

5-14-2563-2 

•0073.V2941 

137 

18769 

2571353 

11-7040999 

5-1551367 

•007299-270 

138 

19044 

5628072 

]]-747:»401 

5-1676493 

•(KI7-240K77 

139 

19321 

2685619 

11-7898261 

5-1801015 

•007194-245 

140 

19600 

2744000 

118321596 

5-  J  924941 

•00714-2857 

141 

19881 

2803221 

11-8743421 

5-2048279 

•00709-2199 

142 

20164 

2863288 

11-9163753 

5-2171034 

•00704-2-254 

143 

20449 

2924207 

11  -9582607 

5-2293215 

•006993007 

144 

20736 

2985984 

]  2-0000000 

5-24148-28 

•006944444 

145 

21025 

3048625 

1-2-0415946 

5-2535879 

•00089655-2 

140 

21316 

3112136 

1-2  0830460 

5-2656374 

•006849315 

147 

21609 

3176523 

12-1243557 

5-2776321 

•006802721 

148 

2191)4 

3241792 

12  1655251 

5-2895725 

•006756757 

149 

22201 

'   3307949 

122065556 

53014592 

•006711409 

150 

22500 

3375000 

12-2474487 

5-3132928 

•00666!  5fi(!7 

151 

22801 

3442951 

12-2882057 

5-3250740 

•0006'2-25I7 

152 

23104 

351  1808 

123288280 

5-3368033 

•006578947 

153 

23409 

3581577 

]  2-36931  69 

5-3484812 

•000535948 

154 

23716 

36522(54 

12-4096736 

5-3601084 

•00649351  H> 

155 

24025 

3723875 

12-449H996 

5-3716854 

•006451613 

156 

24336 

3796416 

12-4899960 

5-3832126 

•006410-250 

157 

24649 

3869893 

l-2-5-29'.)64l 

5-3946907 

•0003094-27 

158 

24964 

3944312 

125698051 

5-4061-20-2 

•00(5329114 

159 

25281 

4019679 

12-6095202 

5-4175015 

•00(5-289308 

160 

25600 

4096000 

1-26491106 

54288352 

•1)06-250000 

161 

25921 

4173281 

1-2-6^85775 

5-4401218 

•006-211180 

102 

26244 

4251528 

12-72792-21 

5-451  36  18 

•00617-2840 

163 

26569 

4330747 

12-7671453 

54625556 

•006134969 

164 

26896 

4410944 

12-8062485 

5-4737037 

•006097561 

165 

27225 

4492125 

12-8452326 

5-4848066 

•006060606 

166 

27556 

4574296 

]  2-8840987 

5-4958647 

•00(5024096 

167 

27889 

4657463 

12-9228480 

5-5068784 

•OO.YJS8024 

168 

28224 

4741632 

129614814 

5-5178484 

•00595-2381 

169 

28561 

4826809 

13-0000000 

5-5287748 

•005917160 

170 

28900 

4913000 

13-0:{84048 

5-5396583 

•005882:553 

171 

29241 

5000211 

130766968 

5-55049IH 

•005817(153 

172 

29.584 

5088448 

131148770 

5-561-2978 

•005813953 

173 

299'29 

5177717 

13-15-2<)404 

5-5720546 

•0057H0347 

174 

30276 

5268024 

13-19(19060 

5-582770-2 

•005747126 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


105 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

175 

30025 

5359375 

13-2287560 

55934447 

005714286 

170 

30976 

5451770 

13-2664992 

5-6040787 

•005081818 

177 

31329 

5545233 

13-3041347 

5-6146724 

•0050497  !R 

178 

31084 

5639752 

133416641 

56252203 

•0056I797R 

179 

32041 

5735339 

13-3790882 

5-6357408 

•00558055)2 

180 

32400 

5832000 

13-4164079 

50402102 

•005555556 

181 

32761 

5929741 

13-4536240 

5-0500528 

•005524862 

182 

33124 

0028568 

13-4907376 

5-0670511 

•005494505 

183 

334  H9 

6128487 

135277493 

5-07741  14 

•005464481 

184 

33H50 

0229504 

13-5646(500 

5-6877340 

•005434783 

185 

34225 

0331025 

13-0014705 

5-6980192 

•005405405 

186 

34590 

0434856 

13-038  18]  7 

5-7082675 

•005376344 

187 

34969 

0539203 

13-0747943 

5-7184791 

•005347594 

18S 

35344 

6644072 

13-7113092 

5-7280543 

•005319149 

189 

35721 

6751209 

13-7477271 

5-7387936 

•005291005 

]!K) 

36100 

'  6859000 

13-7840488 

5-7488971 

•005263  1  58 

191 

36481 

6967871 

13-8202750 

5-7589052 

•005235602 

192 

36864 

7077888 

13-8504065 

5-7  >S9982 

•005208333 

193 

37249 

7189017 

13-8924440 

5-7789906 

•005181347 

194 

37036 

7301384 

13-9283883 

5-7889004 

•005154039 

195 

38025 

7414875 

13-9642400 

5-7988900 

•0051  '28205 

190 

38410 

7529536 

14  -0000000 

5-8087857 

•005102041 

197 

38809 

7645373 

14-0350088 

5-8180479 

•005070142 

198 

39204 

77(52392 

14-0712473 

5-8284767 

•0050.50505 

199 

39001 

7880599 

H'1067360 

5-8382725 

•005025126 

200 

40000 

8000000 

H'1421356 

58480355 

•005000000 

20  J 

40401 

8120601 

14'17744G9 

5-8577600 

•004975124 

202 

40804 

8242408 

14-2120704 

5-8074043 

•004950495 

2o:i 

41209 

8365427 

14-2478008 

5-8771307 

•004926108 

201 

41010 

8489004 

14-2828509 

58807653 

•004901961 

20.") 

42025 

8GJ5125 

H'317821  1 

5-8903085 

•004878049 

306 

42430 

8741810 

14-3527001 

5-9059406 

•004854369 

207 

42849 

8809743 

143874946 

5-9154817 

•004830918 

208 

43264 

8998912 

14-4222051 

5-9249921 

•0048070!)2 

209 

43081 

9129329 

14-4508323 

5-9344721 

•00478408!) 

210 

44100 

9201000 

14-4913707 

5-9439220 

•004761905 

2J1 

44521 

9393931 

14-5258390 

5-9533418 

•004739336 

212 

44944 

9528128 

14-5002198 

5-9627320 

•004716981 

21.3 

45369 

.9663597 

14-5945195 

5-9720926 

.  -004  094836 

214 

45790 

9800344 

14-0287388 

5-9814240 

•004672897 

215 

40225 

9938375 

14-6028783 

5-9907264 

•004651163 

210 

46650 

10077696 

14-6969385 

6-0000000 

•004029630 

217 

47089 

10218313 

147309199 

6-0092450 

•004608295 

218 

47524 

10360232 

14-7648231 

6-0184617 

•004587156 

219 

47961 

10503459 

14-7986486 

6-0276502 

•004566210 

220 

48400 

10648000 

148323970 

6-0368107 

•004545455 

221 

48841 

10793861 

14-8660687 

60459435 

•004524887 

2-22 

49284 

10941048 

14-8996644 

6-0550489 

•004504505 

2-23 

49729 

11089567 

14-9331845 

6-0641270 

•004484305 

224 

50176 

1J  239424 

14-9666295 

6-0731779 

•004404286 

225 

50025 

1  1390625 

15-OuOOOOO 

6-0822020 

•004444444 

226 

51070 

11543176 

15-0332904 

6-0911994 

•004424779 

227 

51529 

]J  097083 

15-0065192 

6-1001702 

•004405286 

228 

51984 

1  1852352 

150996689 

6-1091147 

•004385905 

22') 

52441 

12008989 

15-1327460 

6-1180332 

•00430(5812 

230 

52900 

12167000 

15-  1057509 

6-1269257 

•004347826 

231 

53361 

12320391 

15-  J  986842 

6-1357924 

•0043-29004 

23-2 

53824 

12487168  • 

152315402 

6-1446337 

•004310345 

233 

54289 

121549337 

15-2643375 

6-1534495 

•004291845 

234 

54756 

12812904 

15-2970585 

6-1622401 

•004273504 

335 

55225 

12977875 

15-3297097 

6-1710058 

•004255319 

230     55090 

13144256 

15-3622915 

6-1797466 

•004237288 

166 


SQUAKES,  CUBES,  ETC.,  OF  NUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocal*. 

237 

56169 

13312053 

15-3948043 

6-1884628 

•004219409 

238 

50(544 

13481272 

15-4272486 

6-1971544 

•004201681 

2:39 

57121 

13651919 

15-4596248 

6-2058218 

•004184100 

240 

57600 

13824000 

15-4919334 

6-2144(55;) 

•004  10(5607 

241 

58081 

13997521 

1  5-  524  J  747 

6-2230843 

•004141(378 

212 

585(54 

141724^ 

15-5563492 

6-2316797 

•0041322:!! 

213 

59049 

14348907 

1  5-58*1573 

6-2402515 

•004115226 

244 

59536 

1452(5784 

15-  (5204994 

6-24879D3 

•00409^3(51 

245 

60025 

14706125 

15-6524758 

6-2573248 

•004081633 

246 

60516 

1488693(5 

15-6843871 

6-2(i582(56 

•0040(55041 

247 

61009 

15069223 

157102336 

6-2743054 

•004048583 

248 

61504 

15252992 

15-7480157 

6-2827613 

•004032258 

249 

62001 

15438249 

15-7797338 

6-2911946 

•00401(50(54 

250 

62500 

15625000 

15-8113883 

6-299(5053 

•IKMOOOOOO 

251 

63001 

15813251 

15-8429795 

6-307HK35 

•0039840(54 

25-2 

63504 

16003008 

15-874.5079 

6-31(535915 

•00390H254 

253 

64009 

.16  194277 

15-9059737 

6-3247035 

•0039525(59 

254 

64516 

163870(54 

15-9373775 

6-333025(5 

•003937008 

255 

65025 

16581375 

15-9687194 

6-3413257 

•0039215(59 

256 

65536 

16777216 

16-0000000 

6-3496042 

•00390(5250 

257 

66049 

1697451)3 

16-0312195 

6-3578611 

•003891051 

258 

66564 

17173512 

JO-0623784 

6-3660968 

•0038759(59 

259 

67081 

17373979 

16-0934709 

6-3743111 

•0038(51004 

260 

67(500 

17576000 

16-1245155 

6-3825043 

•00384(5154 

261 

68121 

17779581 

16-15.54944 

6-39067(55 

•003831418 

262 

68644 

17984728 

16-18(54141 

6-3988279 

•00381(5794 

203 

69169 

18191447 

162172747 

6-40(59585 

•003H02281 

264 

69696 

18399744 

16-24807(58 

6-4150(587 

•003787879 

265 

70225 

18609625 

16-2788206 

6-4231583 

•003773585 

266 

70756 

18821096 

16-3095064 

6-431227(5 

•003759398 

267 

71289 

19034163 

1(5-3401346 

6-4392767 

•003745318 

268 

71824 

19248832 

16-3707055 

6-4473057 

•003731343 

2(59 

72361 

19465109 

16-4012195 

6-4553148 

•003717472 

270 

721)00 

19(583000 

16-4316767 

6-4(533041 

•003703704 

271 

73441 

19902511 

16-4620776 

6-4712736 

•003(590037 

272 

73!)84 

20123(548 

16-4924225 

6-4792236 

•003676471 

273 

74529 

2034(5417 

16-5227116 

6-4871.541 

•003663004 

274 

75076 

20570824 

16-5529454 

6-4950653 

•003649(535 

275 

75625 

20796875 

16-5831240 

6-5029572 

•003(531)3(54 

276 

76176 

2102457(5 

16-6132477 

6-5108300 

•003(523188 

277 

76729 

21253933 

16-6433170 

6-5186839 

•003610108 

278 

77284 

21484952 

16-673332) 

6-5265189 

•00359712-2 

279 

77841 

21717639 

16-7032931 

8"  53  ^  1351 

•003584229 

280 

78400 

2195-2000 

16-7332005 

6-5421326 

•003571429 

281 

78961 

22188041 

16-7630546 

6-5499116 

•003558719 

282 

79524 

22425768 

16-7928556 

6-5576722 

•003541)099 

283 

80089 

22(565187 

16-822(5038 

6-5654144 

•0035335(59 

284 

80656 

22906304 

16-8522995 

6-5731385 

•003522127 

285 

81225 

23149125 

168819430 

6-5808443 

•003508772 

286 

81796 

23393656 

1(5-91  15345 

6-5885323 

•003496503 

287 

82369 

23(539903 

16-9410743 

6-59(52023 

•003484321 

288 

82944 

23887872 

16-9705627 

6-6038545 

•003472222 

289 

83521 

241375(59 

17-0000000 

6-6114890 

•0034(50208 

290 

84100 

24389000 

7-0293804 

6-6191060 

•003448276 

291 

84(581 

24(542171 

7-0587221 

6'62(i7054 

•00343(542(5 

292 

85264 

24897088 

7-0880075 

6-6342874 

•003424058 

293 

85849 

25153757 

7-1172428 

6-6418522 

•0034129(59 

294 

86436 

25412184 

7-1464282 

6-  (5493998 

•003401361 

295 

87025 

7-1755(540 

6-6509302 

•003389831 

296 

87616 

35934339 

17-204(1505 

6-6644437 

•00337H378 

297 

88209 

2(51981(73 

17-2'n(>879 

66719403 

•0033C7003 

298 

88804 

26463592 

17-2626765 

6-6794200 

•0033a5705 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


167 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

2.13 

89101 

26730899 

17-2916165 

6-6868831  • 

•003344482 

300 

90000 

27000000 

17-3205081 

6-6943295 

•003333333 

331 

90001 

27270901 

17-3493516 

6-7017593 

•003322259 

302 

91234 

27.343608 

17-3781472 

6-7091729 

•00331  1258 

3!>3 

91809 

27318127 

17-4368952 

6-7  1(55700 

•003301330 

304 

92416 

2S094464 

17-4355958 

6-72.59508 

•003289474 

305 

93025 

28372625 

17-4642492 

6-7313155 

•003278689 

3015 

93G36 

28G52G16 

17-4928557 

6-7386641 

•0032(57974 

307 

94249 

28934443 

17-5214155 

6-7459967 

•003257329 

338 

94804 

2D218112 

17-5499288 

6-7533134 

•003246753 

309 

95481 

29503G29 

17-5783958 

6-7606143 

•0032,36246 

310 

961  00 

29791000 

17-6068169 

6-7678995 

•003225806 

311 

9G721 

30080231 

17-6351921 

6-7751690 

•003215434 

312 

97344 

30371328 

17-6635217 

6-7824229 

•003205128 

311 

97969 

30664297 

17-6918060 

6-7896613 

•003194888 

314 

98596 

30959144 

17-7200451 

6  7968844 

•003184713 

3J5 

99225 

31255875- 

17-7482393 

6-8040921 

•003174603 

316 

99856 

31554496 

17-7763883 

6-8112847 

•003164557 

31? 

100489 

31855013 

17-8044938 

6-8184620 

•0031.54574 

318 

101124 

32157432 

7-8325545 

G-8250242 

•003144654 

319 

101761 

32461759 

7-8605711 

6-8327714 

•003134796 

320 

102400 

327G8000 

7-8885438 

6-8399037 

•003125000 

321 

103041 

33076161 

7-9164729 

6-8470213 

•003115265 

3-2-2 

103684 

33386248 

7-9443584 

6-8541240 

•003105590 

3-23 

104329 

33698267 

7-9722008 

6-8612120 

•003095975 

3-24 

104976 

34012224 

18-0000000 

6-8682855 

•00308G420 

325 

105625 

34328125 

18-0277564 

6-8753413 

•003076923 

32<J 

106276 

34645976 

18-0554701 

6-8823888 

003067485 

327 

106929 

349G5783 

18-0831413 

6-8894188 

•003058104 

32H 

107584 

35287552 

18-1107703 

6-8964345 

•003048780 

329 

108241 

3561  1289 

18-1383571 

6-9034359 

•003039514 

330 

108900 

35937000 

18-1659021 

6-9104232 

•003030303 

331 

109561 

3G264691 

18-1934054 

6-9173964 

•003021148 

332 

110224 

36594368 

18-2208672 

6-9243556 

•003012048 

333 

110889 

36926037 

18-2482876 

6-9313008 

•003003003 

334 

'111556 

37259704 

18-2756669 

6-9382321 

•002994012 

335 

112225 

37595375 

18-3030052 

6-9451496 

•002985075 

330 

112896 

37933056 

18-3303028 

6-9520533 

•002976190 

337 

113569 

38272753 

18-3575598 

6-9589434 

•002967359 

338 

114244 

38614472 

18-3847763 

6-9658198 

•002958580 

339 

114921 

38958219 

18-4119526 

6-972(5826 

•002949853 

340 

115600 

39304000 

18-4390889 

6-9795321 

•002941176 

341 

116281 

39G51821 

18  466ia53 

6-9863(581 

•002932551 

342 

1  169fV4 

40001688 

18-4932420 

6-<)931906 

•002923977 

343 

117649 

40353607 

IH-5202592 

7-0000000 

•002915452 

344 

118336 

40707584 

18-5472370 

7-00679(52 

•002906977 

345 

119-125 

41063625 

18-5741756 

7-0135791 

•002898551 

346 

119716 

41421736 

18-0010752 

7-0203490 

•002890173 

347 

120409 

41781923 

18-6279360 

7-0271058 

•002881844 

348 

121104 

•  42144192 

18-0547581 

7-033*497 

•002873563 

34P. 

121801 

42508549 

18-6815417 

7-0405806 

•0028(55330 

350 

122500 

42875000 

18-7082869 

7-0472987 

•002857143 

351 

123201 

43243551 

18-7349940 

7*0540041 

•002849003 

352 

123904 

43614208 

18-7616630 

7-0606967 

•002840909 

353 

124609 

43986977 

18-7882942 

7-0673767  - 

•002832801 

354 

125316 

44361864 

18-8148877 

7-0740440 

•002824859 

355 

12(5025 

44738875 

18-8414437 

7-0806988 

•002816901 

35(i 

12(5736 

45118016 

18-8679023 

7-0873411 

•002808989 

357 

127449 

45499293 

18-8944436 

7-0939709 

•002801120 

358 

128164 

45882712 

18-9208879 

7-1005885 

•002793296 

359 

128881 

4(5268279 

18-9472953 

7-1071937 

•002785515 

360 

129600 

46656000 

18-9736660 

7-1137866 

•002777778 

168 


SQUARES,  CUBES,  ETC.,  OF 


No. 


Squares. 


Square  Roots.          Cube  Roots. 


Reciprocal* 


361 

130321 

47045881 

19-0000000 

71203674 

•002770083 

362 

131044 

47437928 

190202976 

7-  J  269360 

•002762431 

363 

131769 

47832147 

19-0525589 

7-  1334!  125 

•002754821 

364 

1324% 

482-28544 

190787840 

7-1400370 

•00-2747-253 

365 

133225 

48627125 

191049732 

71465695 

•00-2739726 

366 

133956 

49027896 

191311265 

7-1530901 

•002732-240 

367 

134689 

49430863 

19  1572441 

7-1595988 

•002724796 

368 

135424 

49836032 

19  1833261 

7-1660957 

•002717391 

363 

136161 

50243409 

192093727 

7  1725809 

•002710027 

370 

136900 

50653000 

192353841 

7-1790544 

•00-270-2703 

371 

137641 

51064811 

192613603 

7-1855162 

•002695418 

372 

138I584 

51478848 

192873015 

7-]  9  19663 

•00-2688172 

373 

139129 

51895J17 

193132679 

7-1984050 

•00-2080905 

374 

139876 

52313624 

19  3390796 

7  2048322 

•002673797 

375 

140625 

52734375 

193649167 

7-2112479 

•002066607 

376 

141376 

53157376 

19  3907194 

7-2176522 

•002(559574 

377 

142129 

53582633 

194164878  • 

7-2240450 

•002(552520 

378 

142884 

54010152 

194422221 

7-2304268 

•01)2645503 

379 

143641 

54439939 

194679223 

7-2367972 

•00<-263852l 

380 

144400 

M872000 

19493588~ 

72431565 

•002631579 

381 

145181 

55306341 

195192213 

72495045 

•002624672 

382 

145924 

55742968 

19.1448203 

72558415 

•002017801 

383 

140689 

56181887 

195703858 

7-2621675 

•OOC6  10906 

384 

147456 

56623104 

195959179 

7-2684824 

•002604167 

385 

148225 

57066625 

19  6214109 

7-2747864 

•002597403 

386 

148996 

57512456 

19  6468,827 

7-2810794 

•002590674 

387 

149769 

57960603 

196723156 

7-2873617 

•002583979 

388 

150544 

58411072 

196977156 

7-2936330 

•002577320 

389 

151321 

58863869 

19-7230829 

72998936 

•002570694 

390 

152100 

59319000 

197484177 

7-3061436 

•0025(54103 

391 

153881 

5U776471 

19  7737199 

7-3123828 

•00-2557545 

392 

153664 

60236288 

197989899 

7-3186114 

•00-255  1  0-20 

3!  >3 

154449 

60698457 

19  8242276 

7-3348295 

•0025445-29 

394 

155236 

61162984 

19  84!)4:532 

7-3310369 

•002538071 

395 

156025 

61629875 

19  8746009 

7-3372339 

•002531(546 

396 

156816 

62099136 

19-8997487 

7-3434205 

•002525253 

397 

157609 

62570773 

19-9248588 

7-3495966 

•002518892 

398 

158404 

63044792 

199499373 

7-3557624 

•002512563 

L99 

159201 

63521199 

19  9749844 

7-3619178 

•002506266 

400 

IWOOO 

64000(100 

200000000 

7-3680(530 

•002500000 

401 

160801 

64481201 

20  0249844 

7-3741979 

•002493706 

402 

161604 

04964808 

200499377 

7-3803-227 

•00-24875(52 

403 

162409 

65450827 

20  0748599 

7-3864:173 

•00-2481390 

404 

163216 

65939264 

20  09975  J  2 

7-3925418 

•002475248 

405 

164025 

66430125 

201246118 

7-3986363 

•0024(59136 

406 

164836 

66923416 

201494417 

7-4047206 

•002403054 

407 

165649 

67419143 

20  1742410 

7-4107950 

•002457002 

408 

166464 

67917312 

201990099 

7-4168595 

•002450980 

409 

167281 

68417929 

20  2-2:57484 

7-4229142 

•00-2444988 

410 

168100 

68921000 

20  2484567 

7-4289589 

•00-2439024 

411 

168921 

69426531 

202731349 

7-4349938 

•002433090 

412 

-  169744 

6<l934.">-28 

202977831 

74410189 

•00-24-27184 

413 

170569 

70444997 

203224014 

74470342 

•002121308 

4J4 

171396 

70957944 

203469899 

74530399 

•002415459 

415 

172225  !    71473375 

20-3715488 

7  4590:i;>9 

•002409639 

416 

173056 

71991296 

20-3960781 

7-4650223 

•002403846 

417 

173889 

72511713 

20-4205779 

7-4709991 

•00-23(.»8082 

418 

174724 

73034632 

20-4451  >4H3 

7-4709004 

•002392344 

419 

175561 

73560059 

204694895 

7-4829242 

•002:186035 

420 

176400 

74088000 

20-4939015 

7-4888724 

•602380952 

421 

177241 

74618461 

20-5182845 

7-4948113 

•002375297 

422 

178084 

75151448 

20V-426386 

7-5007406 

•002369668 

SQUARES,   CUBES,  ETC.,  OF  NUMBERS. 


1C9 


Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

178929 

75686967 

20  5669638 

7-5066607 

•002364066 

179776 

76225024 

20-5912603 

7-5125715 

•002358491 

180625 

76765625 

20-6155281 

7-5184730 

•002352941 

181476 

77308776 

20-6397674 

7-5243652 

•002347410 

182329 

77854483 

20-6639783 

7-5302482 

•002341920 

18:J184 

78402752 

20-6881609 

7-5361221 

•082336449 

184041 

78953589 

20-7123152 

7-5419867 

•002331002 

184900 

79507000 

211-7364414 

7-5478423 

•002325581 

185761 

80062991 

207605395 

7-5536888 

00232018(5 

186624 

80621568 

207846097 

7-5595263 

002314815 

187489 

81182737 

20-8086520 

7-5653548 

•0023094(59 

188356 

81746504 

20-8326667 

75711743 

•002304147 

189225 

82312875 

208566536 

7-5769849 

•002298s.)! 

190096 

82881856 

208806130 

7-5827865 

•002293578 

190969 

83453453 

209045450 

7-5885793 

•002288330 

191844 

84027672 

20-9284495 

7-5943(533 

•002283105 

192721 

84604519 

209523268 

7-6001385 

•002277904 

193600 

85184000 

20-9761770 

7-6059049 

•002272727 

194481 

85766121 

21-0000000 

7-6116626 

•002267574 

1  95364 

86350888 

21-0237960 

7-6174116 

•002262443 

196249 

86938307 

210475652 

7-6231519 

•0022.17336 

197136 

87528384 

210713075 

7-628P837 

•0022.rJ-.232 

198025 

88121125 

210950231 

7-6346067 

•0(12247191 

198916 

88716536 

21-1187121 

7-6403213 

002242152 

199809 

89314623 

21-1423745 

7-6460272 

•002237136 

200704 

89915392 

21-1660105 

7-6517247 

•002232143 

201601 

90518849 

21-1896201 

7-6574138 

002227171 

202500 

91125000 

21  2132034 

7-6630943 

•002222222 

203401 

91733851 

212367606 

7-6687665 

•002217295 

204304 

92345408 

21-2602916 

7-6744303 

002212389 

205209 

92959677 

21-2837967 

7-6800857 

•002207506 

206116 

93576664 

21-3072758 

7-6857328 

•002202643 

207025 

94196375 

21-3307290 

7-6913717 

•002197802 

207936 

94818816 

21-3541565. 

7-6970023 

•002192982 

208849 

95443993 

21-3775583 

77026246 

•002188184 

209764 

96071912 

21-4009346 

7-7082388 

•0021834015 

210681 

96702579 

214242853 

7-7188448 

•002178649 

211600 

97336000 

214476106 

7-7194426 

•002173913 

212521 

97972181 

21-4709106 

7  7250325 

•002169197 

213444 

9P61112S 

21-4941853 

77306141 

•0021  64502 

214369 

99252847 

21-5174348 

7-7361877 

•002159827 

215296 

99897344 

21-540C592 

7-7417532 

•002155172 

216225 

100544625 

21-5638587 

7-7473109 

•002150538 

217156 

101J94696 

21-5870331 

7-7528606 

•0021451)23 

218089 

101847563 

21-6101828 

7-7584023 

•002141328 

219024 

102503232 

21-6333077 

7-7639361 

002136752 

219961 

103161709 

21-6564078 

7-7694620 

002132196 

220900 

103823000 

21-6794834 

7-7749801 

-002127660 

221841 

104487111 

21-7025344 

7-7804904 

•00212:5142 

222784 
223729 

105154048 

105823817 

21-7485632 

7-7859928 
7  7914875 

•002118644 
'002114165 

224676 

10C.496424 

217715411 

7-7969745 

002109705 

225625 

107171875 

21-7944947 

7-8024538 

002105263 

226576 

107850176 

218174242 

•7-8079254 

•002100840 

227529 

108531333 

218403297 

78133892 

0020964H6 

228484 

109215352 

21  8632111 

7-8188456 

•002092050 

229441 

109902239 

21-8860686 

7-8242942 

002087(583 

230400 

110592000 

21  91  189023 

78297353 

•002083333 

2313151 

1112841)41 

219317122 

7-8351688 

•002079002 

232324 

11I!)*01()8 

21-9544984 

78405949 

•002074689 

2332^9 

11  2678587 

21-9772610 

7  8460134 

•002070393 

234256 

113379904 

22  0000000 

78514244 

•002066116 

170 


SQUARES,  CUBES,  ETC.,  OF  XUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

485 

233225 

1  1  4084  1  25  '  2202271  55 

7-8568-281 

•002061856 

480 

23G196 

114791256 

28-0454077 

7-862-2-242 

•002057613 

487 

2371153 

115501303 

2-2  OIW0765 

78676130' 

•002053388 

488 

238144 

116214272 

22-0907220 

7-8729944 

•002049180 

483 

23J121 

110930169 

22  U  33444 

7-8783684 

•00-2044990 

490 

24J100 

117649000 

22  1359436 

7  8837352 

•002040816 

491 

241081 

118370771 

22-1585198 

7-8890946 

•002030660 

49-2 

2420G4 

119095488 

22-1810730 

7-8944468 

•002(1325-20 

4U3 

243049 

119823157 

22  2036033 

7-8997917 

•0020-28398 

494 

24403;  > 

120553784 

222261108 

7-905  12H4 

•002024-291 

495 

245025 

121287375 

222485953 

79104599 

•00-20-20202 

496 

246016 

122023936 

222710575 

79157832 

•002016129 

497 

247009 

122763473 

22-2934968 

79210994 

•002012072 

498 

248004 

123505992 

223159136 

7-9264085 

•00-200803-2 

499 

249001 

124251499 

223383079 

7-9317104 

•002004008 

500 

25iK)00 

125000000 

223606798 

79370053 

•002000000 

501 

251001 

125751501 

223830293 

79422931 

•00199(5008 

502 

252004 

126506008 

224053565 

7-9475739 

•Oi)19il-2032 

503 

253009 

127263527 

22-4276615 

79528477 

•001988072 

504 

254016 

128024064 

224499443 

7-9581144 

•001984127 

503 

255025 

128787625 

224722051 

7-9633743 

•001980198 

506 

256036 

123554216 

22  4J44438 

7-9686271 

•0019702^5 

507 

257049 

130323843 

225166605 

7-9738731 

001972387 

508 

258064 

131096512 

225388553 

7-9791122 

•001908504 

503 

259081 

131872229 

225610283 

7-9843444 

•001964637 

510 

260100 

132651000 

225831796 

7-9895637 

•001960785 

511 

261121 

133432831 

22-6053091 

7-9947883 

•001956947 

512 

262144 

134217728 

22-6274170 

8-0000000 

•001953125 

513 

263169 

135005697 

22-6495033 

8-0052049 

•001949318 

514 

264196 

135796744 

22-6715631 

80104032 

•001945525 

513 

265225 

136590875 

226936114 

80155946 

•001941748 

516 

266256 

137383096 

227153334 

80207794 

•001937984 

517 

267289 

138188413 

227376341 

8  0259574 

•001934236 

518 

208324 

133991832 

22-7596134 

80311287 

•001930502 

519 

269361 

139798359 

22-7815715 

80362935 

•001926782 

520 

270400 

1401308000 

228035085 

80414515 

•001923077 

521 

271411 

141420761 

228-254244 

80466030 

•001919386 

522 

272484 

142236648 

228473193 

80517479 

•001915709 

523 

273529 

143055667 

228691933 

8-0568862 

001912046 

524 

274576 

143877824 

228910463 

80620180 

•001903397 

525 

275625 

144703125 

22  9128785 

80671432 

•001904762 

526 

276676 

145531576 

22  9346899 

80722620 

•001901141 

527 

277729 

146363183 

22-9564806 

80773743 

•001897533 

523 

278784 

147197952 

22-9782506 

80824800 

•001893939 

529 

270341 

148035389 

230000000 

80875794 

•001890359 

530 

233900 

148877001 

230217289 

8-0926723 

•001H80792 

531 

281331 

149721291 

230434372 

8-09775P9 

•001883239 

532 

283024 

150568763 

230651252 

8-  1023390 

•001879699 

533 

284089 

151419437 

23  0867928 

8-1079128 

•001876173 

531 

285156 

152273304 

23-1084400 

8-1129803 

•001872059 

535 

283225 

153130375 

23  1300670 

8-1  1804  H 

•001869150 

53o 

287238 

153393656 

23  151G738 

8-1230902 

•001865672 

537 

2383G9 

154354153 

23-1732605 

fi-  1281  447 

•001H62197 

538 

289444 

155720872 

231948270 

8-1331870 

•001858736 

539 

290521 

15G590819 

23  2163735 

8-  1382230 

•00  J  855288 

54  3 

291600 

157464000 

23  2379001 

8  1432529 

•001851852 

541 

292681 

1:8349421 

21  2594007 

8  14H27F5 

•001843429 

542 

293764 

159220088 

2']  2-^8335 

8-1532939 

•001845018 

543 

294849 

169103007 

2'5  3023694 

8  1583051 

•00184  JH21 

544 

295936 

160989184 

2<V323837f> 

8  103310-2 

•001Kl4J-j:!5 

545 

297025 

'  161878625 

23  3-152  '51 

8-K;83092 

•001H34M»ia 

541. 

298113 

1G277133G 

23  3666429 

81733020 

001831503 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


171 


" 

Scuares. 

Cubes. 

Square  Roots. 

Cube  Roots, 

Reciprccals. 

47 

299209 

1636(57323 

23-3830311 

8-1782888 

•001828154 

48 

300304 

164566592 

23-4093998 

8-1832695 

•001824818 

49 

301401 

165469149 

23-4307490 

8-1882441 

*  -001821494 

so 

302500 

166375000 

23-4520788 

8-1932127 

•00181818-2 

11 

303601 

167284151 

23-4733892 

8-1981753 

•001814H82 

V2 

304704 

168196608 

23-4946802 

8-2031319 

•001811594 

53 

305809 

169112377 

23-5159520 

8-2080825 

•001808318 

54 

306916 

170031464 

23-5372046 

8-2130271 

•001805054 

55 

308025 

170953875 

23-5584380 

8-2179657 

•001801802 

56 

309136 

171879616 

23-5796522 

8-2228985 

•001798561 

r>7 

310249 

172808693 

23-6008474 

8-2278254 

•001795332 

58 

311364 

173741112 

23-6220236 

8-2327463 

•001792115 

59 

312481 

174676879 

23-6431808 

8-2376614 

•00(788901) 

(50 

313600 

175616000 

23-6643191 

8-2425706 

•001785714 

61 

314721 

176558481 

23-6854386 

8-2474740 

•001782.531 

r,2 

315844 

177504328 

23-7065392 

8-2523715 

•00  77J359 

53 

316969 

178453547 

23-7276210 

8-2572633 

•001776199 

(54 

318096 

179406144 

23-7486842 

8-2621492 

•001773050 

(15 

319225 

180362125 

23-7697286 

8-2670294 

•001769912 

66 

320356 

181321496 

23-7907545 

8-2719039 

•001766784 

87 

321489 

182284263 

23-8117618 

8-2767726 

•001763668 

68 

322624 

183250432 

23-8327506 

8-2816355 

•0017(50563 

69 

323761 

184220009 

23-8537209 

8-2864928 

•001757461) 

70 

324900 

185  J  93000 

23-8746728 

8-2913444 

•001754386 

71 

326041 

186169411 

23-8956063 

8-2961903 

•0-,)  175  13  13 

7-2 

327184 

187149248 

23-9165215 

8-3010304 

•00174d253 

715 

328329 

188132517 

23-9374184 

8-3058651 

00  17452  Jl 

74 

329476 

1891  19224 

23-9582971 

8-3106941 

•OOJ7421M) 

7.1 

330625 

I9010'.)375 

23-9791576 

8-3155175 

•00  1  73  J  130 

7ti 

331776 

1911021)76 

24-0000000 

8-3203353 

•001730111 

77 

332929 

192100033 

24-0208243 

8-3251475 

•001733102 

78 

334084 

193100552 

24-0416306 

8-3299542 

•001730104 

79 

33.1241 

194104539 

24-0624188 

8-3347553 

•001727116 

HO 

336400 

195112000 

24-0831891 

8-3395509 

•001724138 

81 

337561 

196122941 

24-1039416 

8-3443410 

•001721170 

83 

338724 

11)7137368 

24-1246762 

8-3491256 

•001718213 

.H3 

339889 

198155287 

24-1453929 

8-3539047 

•001715266 

84 

341056 

199176704 

24-1660919 

8-3580784 

•001712329 

85 

342225 

200201625 

24-1807732 

8-3634466 

•00.703402 

>8t; 

343390 

201230056 

24-2074369 

8-3682095 

•001706485 

>87 

3445151) 

202262003 

24-2280829 

8-3729668 

•001703578 

>88 

345744 

2032J7472 

24-2487113 

8-3777188 

•001700680 

>8<) 

346921 

204336469 

24-2693222 

8-3824653 

•0016117793 

)tl!) 

348100 

205379000 

24-2899156 

8-3872065 

•001694915 

><U 

341)281 

206425071 

24-3104996 

8-3919423 

•001692047 

>!I2 

350464 

207474688 

24-3310501 

8-3966729 

•001681)189 

393 

351649 

208527857 

24-3515913 

8-4013981 

•001686341 

>9i 

352836 

209584584 

24-3721152 

8-4061180 

•OJKi835')2 

ill.> 

354025 

210644875 

24-3926218 

8-4108326 

•00168,0672 

V.M; 

355216 

211708736 

24-4131112 

8-4155419 

•001677652 

3!)7 

356409 

212776173 

24-4335834 

8-4202460 

•001675042 

>i)8 

357604 

213847192 

24-4540385 

8-4249448 

•001672241 

39'J 

358801 

214921799 

24-4744765 

8-4296383 

•001669449 

tilMJ 

360000 

216000000 

24-4948974 

"8-4343267 

•00166(5667 

Jill 

361201 

217081801 

24-5153013 

8-4390098 

•0016G38J4 

BU2 

3(52404 

218167208 

24-5356883 

8-4436877 

•001661130 

>;)3 

363609 

211)256227 

24-5560583 

8-4483605 

•001058.T75 

504 

364816 

220348864 

24-5764115 

8-4530281 

•001655621) 

G05 

300025 

221445125 

24-5967478 

8-4576906 

•001652893 

500 

367236 

22-25451)16 

24-6170673 

8-4623479 

•001650165 

507 

368449 

223648543 

24-0373700 

8-4670001 

•001647446 

60d 

369664 

224755712 

24-6576560 

8-4716471 

•001644737 

172 


SQl'AKES,  CUBES,  ETC.,  OF  NUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

609 

370881 

225806529 

24-0779254 

8-4702892 

•001042036 

till) 

372100 

220981  (MM) 

24-09X1781 

8-4809201 

•001039344 

Gil 

373321 

228099)31 

24-7184142 

8-4855579 

•001031)001 

613 

374544 

229220-128 

25-73803:58 

8-4901848 

•00  10331)87 

613 

375765) 

23104635)7 

24-7588308 

8-49-18065 

•0016:11321 

C,I4 

3705)96 

231475544 

24-775)0234 

8-45)514233 

•001028004 

615 

378225 

232008375 

24-7991935 

8-5040350 

•00  1  0201  U  6 

(il(i 

379456 

233744890 

24-815)3473 

8-5080417 

•001023377 

017 

380089 

234885113 

24-8394847 

8-5132435 

•001620740 

Gis 

381924 

23!il)2lM)3i 

24-8590058 

8-5178403 

•001(518123 

619 

383101 

237170059 

24-8797106 

8-5224331 

•001615509 

620 

384400 

238328(100 

24-85)97992 

8-5270189 

•001612903 

021 

38nd41 

239483061 

24-9198716 

5-53  10009 

•001010300 

022 

38U8H4 

240041848 

24-9399278 

85301780 

•001007717 

023 

388129 

24  J  804  307 

24-951)9079 

8-5407501 

•00100513(5 

624 

38i)37G 

242970024 

24-9791)920 

&5453173 

•001602504 

625 

390625 

244140025 

25-0000000 

8-5498797 

•001000000 

020 

391876 

2453  1  4376 

23-0  101)920 

8-5544372 

•001597444 

027 

393129 

240491883 

25-035)9081 

8-558985)9 

•001594896 

628 

394384 

247073152 

25-059.I282 

8-5035377 

•001592357 

629 

3S»5«>41 

248858189 

25-0798724 

8-5080807 

•0015811  182.', 

(iiill 

390901) 

250047000 

25-0998008 

8-5720189 

•001587302 

631 

398101 

251239591 

25-1197134 

8-5771523 

•001584786 

032 

3911424 

252435908 

25-135)0102 

8-5810809 

•001582278 

633 

400089 

253030137 

25-155)4913 

8-5802047 

•001579779 

G34 

401956 

254840104 

25  179356G 

8-5907238 

•001577287 

635 

403225 

250047875 

25  1992003 

8-5952380 

•001574803 

636 

404490 

257259456 

25-215)0404 

8-5997476 

•001572327 

G37 

405709 

258474853 

25-2388.->89 

8-0042525 

•0015(59859 

638 

407044 

2591)94072 

252580019 

8-6887528 

•001507398 

639 

408321 

2605)17119 

25-2784493 

8-0132480 

•001504945 

640 

409000 

202)44000 

25-2982213 

8-0177388 

•0015(52500 

641 

410881 

203374721 

25-3179778 

8-0222248 

•001500062 

642 

4J2I04 

2(54009288 

25-3377189 

8-02070(53 

•001557032 

643 

413449 

205847707 

2.V  3574447 

8-0311830 

•001555210 

644 

414736 

2670895)84 

25-3771551 

8-0356551 

•001552795 

645 

41GI25 

268336125 

25-3908502 

8-G40122!) 

•001550388 

64ii 

417316 

369583136 

25-4105302 

8-0445855 

•001547988 

647 

418009 

270840023 

25-4301947 

8-6490437 

•001545595 

648 

419904 

272097792 

25-4558441 

8-0534974 

•001543210 

649 

421201 

273359449 

25-4754784 

8-0579405 

•001540832 

650 

422500 

274025000 

25-4950976 

8-6(523911 

•ooi  :>:!:<  -102 

651 

423801 

2758:14451 

25-5147016 

8-o:;o83io 

•001536098 

652 

425104 

277167808 

25-53429(fi 

8-6712(505 

•(KH533742 

653 

420409 

278445077 

25-5538047 

8-6736974 

•00153135)4 

654 

427716 

279720204 

25-5734237 

8-0801237 

•0015291  152 

655 

429025 

281011375 

25-5929678 

8-0845450 

•00152(5718 

656 

430336 

282300416 

25-0124909 

8-6889030 

•00152435)0 

657 

431039 

283593393 

25-0320112 

8-  65  133755) 

•001522070 

658 

432904 

284890312 

2.VG5I5107 

8-(  5!  177843 

•001519757 

659 

434281 

28fil91179 

25-07095)53 

8-7021882 

•00151/451 

660 

435000 

287490000 

250904052 

8-7005877 

•001515152 

6G1 

4361)21 

288804781 

25-7099203 

8-7109827 

•001512859 

602 

438244 

290117528 

25-725)3007 

8-7153734 

•001510574 

GG3 

439509 

25)1434247 

25-7487804 

8-7197590 

•00150825)6 

604 

440896 

292754944 

25-7081975 

87241414 

•00150(5024 

COS 

442225 

25)4079025 

25-78755)35) 

8-7285187 

•001503755) 

606 

44355G 

25)5408290 

25-8005)758 

•8-7328918 

•001501502 

607 

444889 

25)0740903 

25-82(53431 

8-7372004 

•001499250 

608 

440224 

29*077632 

25-8450900 

8-7410240 

•001497000 

609 

447501 

299)18309 

25-865*  134  3 

8-7459840 

.001494708 

670 

448900 

3007G3000 

25'8843582 

8-7503401 

•001492537 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


No.   '   Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

671 

450241 

302111711 

25-9036677 

8-7546913 

•001490313 

672 

451584 

3034(54448 

25-  i  (229628 

8-7590383 

•001488095 

673 

452929 

304821217 

259422435 

8-7633809 

•001485H84 

674 

454276 

306182024 

25-9615100 

8-7677192 

•0014836HO 

675 

455625 

307546875 

25-9807621 

8-7720532 

•00148U81 

676 

456976 

308915776 

26-0000000 

8-7763830 

•OON7W90 

677 

458329 

310288733 

26-0192237 

8-7807084 

•001477i()5 

678 

459684 

311665752 

26-0384331 

8-7850296 

•001474926 

679 

401041 

313046839 

26-0576284 

8-7893466 

•001472754 

680 

4G2400 

314432000 

26-0768096 

8-7936593 

•001470.V8 

681 

463761 

315821241 

26-0959767 

8-7979679 

•00  14(58429 

682 

465124 

317214568 

26-1151297 

8-8022721 

•001466276 

683 

466489 

318611987 

26-1342687 

8-8065722 

•OOH6412!) 

684 

467856 

320013504 

26-1533937 

8-8108681 

•00  146  1!  188 

685 

469225 

321419125 

26-1725047 

8-8151598 

•00145DH.14 

686 

470596 

322828856 

26-1916017 

8-8194474 

•0014577-2(5 

687 

471969 

324242703 

26-2106848 

8-8237307 

•001455604 

688 

473344 

325660672 

26-2237541 

8-8280099 

•001453488 

689 

474721 

327082769 

26-2488095 

8-8322850 

•001451379 

690 

476100 

328509000 

26-2678511 

8-8365559 

•001449275 

691 

477481 

329939371 

26-2868789 

8-8408227 

•001447178 

692 

478864 

331373888 

20-3058929 

8-8450854 

•001445087 

693 

480249 

332812557 

26-3248932 

8-8493440 

•001443001 

694 

481636 

334255384 

26-3438797 

8-8535985 

•001440922 

695 

483025 

335702375 

26-3628527 

8-8578489 

•001438849 

696 

484416 

337153536 

26-3818119 

8-8620952 

•001436782 

697 

485809 

338608873 

26-4007576 

8-8663375 

•001434720 

698 

487204 

340068392 

26-4196896 

8-8705757 

•001432665 

699 

488601 

341532099 

2(5-4386081 

8-8748099 

•001430615 

700 

490000 

343000000 

26-4575131 

8-8790400 

•001428571 

701 

491401 

344472101 

26-4764046 

8-8832661 

•001426534 

702 

492804 

345948408 

26-4952826 

8-8874882 

•001424501 

703 

494209 

347428927 

26-5141472 

8-8917063 

•001422475 

704 

495616 

348913664 

26-5329983 

8-8959204 

•001420455 

705 

497025 

350402625 

26-5518361 

8-9001304 

•001418440 

706 

498436 

351895816 

26-5706605 

8-9043366 

•OOI4Ki4:U 

707 

499849 

353393243 

26-589471(5 

8-9085387 

•001414427 

708 

501264 

354894912 

26-6082694 

8-9127369 

•001412429 

709 

502681 

356400829 

26-6270539 

8-9169311 

•001410437 

710 

504100 

357911000 

26-6458252 

8-9211214 

•001408451 

711 

505521 

359425431 

2(5-6645833 

89253078 

•001406470 

712 

506944 

360944128 

26-6833281 

8-9294902 

•001404494 

713 

508369 

362467097 

26-7020598 

8-9336687 

•001402525 

714 

509796 

363994344 

2(5-7207784 

8-9378433 

•001400560 

7J5 

511225 

365525875 

26-7394839 

8-9420140 

•001398601 

716 

512656 

36706169(5 

26-75817(13 

8-94(51809 

001396648 

717 

514089 

368601813 

26-7768557 

8-9503438  • 

•001394700 

718 

515524 

370146232 

26-7955220 

8-9545029 

•001392758 

719 

516961 

371694959 

26-8141754 

8-9586581 

•001390821 

720 

518400 

373248000 

26-8328157 

8-9628095 

-  -001388889 

721 

519841 

374805361 

26-8514432 

8-9669570 

•001386963 

722 

521284 

371)367048 

26-8700577 

8-9711007 

•001385042 

723 

522729 

377933067 

2(5-8886593 

8-9752406 

•001383126 

724 

524176 

379503424 

2(5-9072481 

8-9793766 

•001381215 

725 

525625 

381078125 

26-9258240 

8-9835089 

•001379310 

726 

527076 

382657176 

26-9443872 

8-987(5373 

•001377410 

727 

528529 

384240583 

26-9629375 

8-9917620 

•001375516 

728 

529984 

385828352 

26-9814751 

8-9958899 

•001373626 

729 

531441 

387420489 

27-0000000 

9-0000000 

•001371742 

730 

532900 

389017000 

27-0185122 

9-0041134 

•0013698(53 

731 

534361 

390617891 

27-0370117 

9-0082229 

•001367989 

732 

5358:11 

392223168 

27-0554985 

9-0123288 

•001366120 

174 


SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roo(s. 

Cube  Roots. 

Reciprocals. 

733 

537289 

393832837 

270739727 

90164309 

•ODI364256 

734 

538756 

395446904 

270924344 

9  0205293 

•0013(52398 

735 

540225 

3117065375 

27-1108834 

9-024(5239 

•0013(50544 

730 

541696 

398(588256 

27-1293199 

90287149 

.•00135869R 

737 

543169 

400315553 

271477439 

90328021 

•CJ1356852 

738 

544644 

401947272 

271661554 

9-03(58857 

001355014 

739 

546121 

403583419 

27-1845544 

9  0409655 

•001353180 

740 

547600 

405224000 

272029410 

90450419 

•001351351 

741 

549081 

40(5869021 

272213152 

90491142 

•00  1341)528 

742 

550564 

408518488 

272396769 

9-0531831 

•001347709 

743 

552049 

410172407 

27-2580263 

9  0572482 

•001345895 

744 

553536 

41  1830784 

27-2763634 

9  0013098 

•001344086 

745 

555025 

413493625 

27-2;)46S81 

9  0653677 

•001342282 

746 

556516 

415160936 

273130006 

9-0694220 

•001340483 

747 

558009 

416832723 

273313007 

9  0734726 

•001338688 

748 

559504 

418508992 

273495887 

90775197 

•00133681)8 

749 

561001 

420189749 

27-3678644 

90815631 

•0013.35113 

750 

562500 

421875000 

27-3861279 

9  0856030 

•001333333 

751 

564001 

423564751 

27-4043792 

9-0896392 

•00133  1558 

752 

565504 

425259008 

274226184 

9-0936719 

•001329787 

753 

567009 

425957777 

274408455 

9  0977010 

•001328021 

754 

568516 

428661064 

27-451)0604 

91017265 

•001326260 

755 

570025 

430368875 

274772633 

9-1057485 

•001324503 

756 

571536 

432081216 

27-4954542 

9-1097669 

•001322751 

757 

573049 

433798093 

275136330 

9-1137818 

•001321004 

758 

574564 

435519512 

27-5317998 

9  1177931 

•001319261 

759 

576081 

437245479 

275499546 

9  1218010 

•001317523 

760 

577600 

438076000 

27-5680975 

91258053 

•001315789 

761 

579121 

440711081 

2758(52284 

9  1298061 

•001314060 

762 

580644 

442450728 

27  6043475 

9  1338034 

•001312336 

763 

582169 

444194947 

276224546 

91377971 

•001310616 

764 

583696 

445943744 

27-6405499 

91417874 

•001308901 

765 

585225 

4471)97125 

27-6586334  - 

91457742 

•001307190 

766 

58675(5 

449455096 

27-(57<57<)r>0 

9-1497576 

•001305483 

767 

588289 

4512J7663 

27-6947648 

9  1537375 

•001303781 

768 

589824 

4.12984832 

27-7128129 

91577139 

•001302083 

769 

591361 

454756609 

277308492 

9  1616869 

•001300390 

770 

592900 

456533000 

27-7488739 

9  1656565 

•001298701 

771 

594441 

458314011 

277668868 

9-1696225 

•0012970  17 

772 

595984 

460099648 

27-7848880 

9  1735852 

•00129.1337 

773 

597529 

461889917 

27  8028775 

91775445 

•00129:>-W51 

774 

'  599076 

463(584824 

278208555 

9  1815003 

•00  129  1990 

775 

6001525 

465484375 

278388218 

91854527 

•00  1290323 

776 

602176 

4672H8576 

2785677(5(5 

9-181)4018 

•001288660 

777 

603729 

4691)97433 

278747197 

9  1933474 

•001287001 

778 

605284 

470910952 

27-8926514 

9  1972897 

•001285347 

779 

606841 

472729139 

27-9105715 

9  2012286 

•001283697 

780 

608400 

474552000 

271)284801 

92051641 

•00  12820:>  1 

781 

609961 

476379541 

27-9463772 

9  209*962 

•001281)410 

782 

611524 

478211768 

27-9IV42629 

92130250 

•001278772 

783 

61308!) 

480048687 

279821372 

92161)50.-) 

•001277139 

784 

614656 

481890304 

28-0000000 

1)  22087215 

001275510 

785 

616255 

483736(525 

280178515 

92247914 

•00127:«85 

786 

61771)6 

48558765(5 

28-0356915 

922S7o«8 

•001272265 

787 

619369 

487443403 

28-0535203 

92326189 

•001270(548 

788 

620944 

489303872 

28(1713377 

9  2365277 

•001269036 

789 

622521 

4911(59069 

28-0891438 

9-2404333 

•0012(57427 

790 

624  100 

41)30391  MH) 

28-10(59386 

92443355 

•0012(55823 

791 

625681 

41)41113671 

28-12-17222 

92482.344 

•001261223 

792 

627624 

49(5793088 

28  1424946 

9-2521300 

•00126-J6-J6 

793 

628849 

498)577257 

28-1(5(12557 

925(50224 

•001261034 

794 

630436 

500566184 

28  1780056 

92599114 

•001259446 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


175 


No. 


Squares. 


Square  Roots.          Cube  Roots. 


Reciprocals. 


795 

632025 

502459875 

28-1957444 

92037973 

•001257862 

790 

6330  l(i 

504358336 

282134720 

9-2(57071)8 

•001256281 

797 

635209 

500201573 

28-2311884 

92715592 

•001254705 

798 

636804 

508169592 

28-2488938 

92754352 

•001253133 

799 

638401 

510082399 

28-2005*81 

92793081 

•001251364 

800 

640000 

5!  2000000 

282842712 

92831777 

•001250000 

801 

641601 

513922401 

283019434 

9-2870444 

001248439 

80-2 

643204 

515849008 

283196045 

9-2909072 

•001246883 

803 

64481)9 

517781627 

283372546 

9294:071 

•001245330 

804 

640416. 

519718464 

283548938 

92986239 

•001243781 

805 

648.125 

521660125 

283725219 

93024775 

•001242236 

800 

649036 

523006616 

28  3901391 

93063278 

001240695 

807 

651249 

52.5557943 

284077454 

93101750 

-001239157 

808 

652804 

527514112 

28  4253408 

9-3140190 

•001237024 

809 

654481 

529475129 

284429253 

9-3178599 

•001236094 

810 

65(5100 

531441000 

284(ii)4989 

93216975 

•001234568 

811 

657721 

533411731 

28-4780017 

93255320 

•00123304G 

812 

659344 

535387328 

28  4956137 

9-3293634 

•001231527 

813 

6609(59 

537367797 

285131549 

9-3331916 

•001230012 

814 

662596 

539353144 

285306852 

9-3370167 

•001228501 

815 

604225 

541343375 

28-5482048 

9-3408386 

•001220994 

810 

665856 

543338496 

285657137 

9-344G575 

•001225490 

817 

607489 

545338513 

285832119 

9-3484731 

•001223990 

8J8 

609124 

547343432 

"280000993 

9-3522857 

•001222494 

819 

670761 

549353259 

286181760 

93560952 

•001221001 

820 

672400 

531368000 

28  6350421 

93599016 

•001219512 

821 

674041 

553387601 

28-6530976 

9-3637049 

•001218027 

822 

675684 

555412248 

286705424 

93675051 

•001216545 

823 

877329 

557441767 

286879766 

93713022 

•001215067 

824 

678976 

559470224 

28-7054002 

93750963 

•001213592 

825 

680625 

501515625 

2-7228132 

9-3788873 

•001212121 

82(5 

682276 

563559976 

287402157 

9-3826752 

•001210654 

827 

683929 

5(55609283 

287576077 

9-3864600 

•001209190 

828 

685584 

567663552 

28-7749891 

9-3902419 

•001207729 

829 

687241 

5(59722789 

28-7923601 

93940206 

•001206273 

830 

688900 

571787000 

28-8097206 

9-3977964 

•001204819 

831 

690561 

573856191 

288270706 

9-4015091 

•001203369 

832 

692224 

575930368 

28-8444102 

9-4053387 

•001201923 

833 

693889 

578009537 

28-8617394 

9-4091054 

•001200480 

834 

695556 

580093704 

28-8790582 

9-4128690 

•001199041 

835 

697225 

582182875 

288963666 

9-4166297 

001197605 

836 

69881)6 

584277056 

289136646 

9-4203873 

001196172 

837 

700569 

586376253 

28  9309523 

94241420 

•001194743 

838 

702244 

588480472 

289482297 

9-4278936 

•001193317 

839 

703921 

590589719 

239654967 

9-4310423 

•001191895 

840 

705600 

592704000 

28  9827535 

9-4353880 

•001190476 

841 

707281 

594823321 

29-0000000 

9-4391307 

•001189061 

842 

708904  • 

596947688 

290.172363 

9-4428704 

•001187648 

843 

710649 

599'.)77107 

290344023 

9-4466072 

•001186240 

844 

712336 

60121  1584 

290516781 

94503410 

•001184834 

845 

714025 

603351125 

2!)  0088837 

94540719 

•001  183432 

846 

715716 

60549.1730 

29  0800791 

9-4577999 

•001182033 

847 

717409 

607(545423 

291032044 

94615249 

•001180638 

848 

719104 

60!KM)1'.»2 

29  1204396 

94652470 

•001179245 

849 

7208ri 

61  1900049 

29  1370046 

9  4089661 

•001177856 

85!) 

722500 

614125000 

29  J547595 

9  4720824 

•001176471 

851 

724201 

610295051 

291719043 

947(539.">7 

•001175088 

852 

7251)04 

618470208 

29  185)0390 

94801001 

•001  173709 

853 

727009 

621)650477 

292001037 

94838136 

•001  172333 

854 

729316 

622835H04 

292232784 

94875182 

•001170960 

855 

731025 

025026375 

292403830 

94912200 

•001169591 

856 

732736 

627222016 

29-2574777 

94949188 

0011G8224 

170 


Squares. 

Cube*. 

Square  Roots. 

Cube  Hoots. 

Reciprocals. 

734449 

629422793 

29-2745(523 

9-498B147 

•001  100H11 

736164 

631028712 

29-2SM6370 

9-5023078 

•(101105501 

737881 

633839779 

29  3087018 

9-5059980 

•OOI104J44 

739600 

630056000 

29-  32575(56 

9-509(5854 

•0011027!)! 

741321 

638277381 

29-3428015 

9-5133699 

•00110i440 

743044 

640503928 

29-3598365 

95170515 

•001  1  000!  13 

744769 

1542735647 

29-3708010 

9-5207303 

•001158749 

746496 

644972544 

29-3938709 

!r  5244003 

•001  157407 

748225 

647214025 

29-510H823 

9-5280794 

•001  150009 

749956 

C49401890 

29-4278779 

9-5317497 

•001154734 

751689 

651714303 

29-4448037 

y-  5354  172 

•001153403 

753424 

653972032 

29-4618397 

9-5390818 

•001152074 

755161 

656234909 

29-4788059 

9-5427437 

•00115U748 

750900 

658503000 

29-4957(524 

9-5464027 

•001  149425 

758641 

6C07703J  1 

29-5127091 

9-55IJ05S9 

•00  1J  48  106 

7003H4 

6C3054848 

29-5296461 

95537123 

•001140789 

702129 

665338617 

29-54(55734 

9-5573630 

•001145475 

76387(5 

667627024 

29-5(534910 

9-5610108 

•001144105 

765625 

669921875 

29-5803989 

9-56-16559 

•001142857 

767376 

672221376 

29-5972972 

9-5682782 

•001141553 

769129 

674526133 

29-61  4  J  858 

9-5719377 

•001140251 

770884 

676836152 

29-6310648 

9  5755745 

•001138952 

772641 

679151439 

29-6479342 

9-5792085 

•001137656 

774400 

681472000 

29  6047939 

9-5828397 

•001130364 

776161 

683797841 

290810442 

9  5864682 

•001135074 

777924 

686128968 

29-6984848 

9-5900939 

•001133787 

779689 

688465387 

29-7153159 

9-5937169 

•001132503 

781456 

690807104 

29-7321375 

9-5973373 

•001131222 

783225 

693154125 

29-7489496 

9-6009548 

•001  129944 

78491)6 

695506456 

29-7657521 

9-6045690 

•001128668 

786769 

697804103 

29-7825452 

9-6081817 

001127396 

788544 

700227073 

29-7993289 

96117911 

•001126126 

790321 

702595369 

29-8161030 

9-6153977 

•001  124859 

792100 

704969000 

29-8328678 

9-6190017 

•001123596 

793881 

707347971 

298496231 

9-6226030 

•001122334 

795664 

709732288 

29-8663690 

9-6262016 

•001121076 

797449 

712121957 

29-8831056 

9  6297975 

•001119821 

799236 

714516984 

29-8998328 

90333907 

•001118568 

801025 

716917375 

29-9165506 

9-0369812 

•001117818 

802816 

719323136 

29-9332591 

9-C405690 

•001116071 

804609 

721734273 

29-9499583 

9-0441542 

•001114827 

806404 

724150792 

29-9666481 

9-6477367 

•001113586 

808201 

726572699 

29-9833287 

9-6513166 

•001112347 

810000 

729000000 

30-0000000 

96548938 

•001  111  111 

811801 

731432701 

30-0166621 

9-8584084 

•001  109878 

813604 

733870808 

30-0333148 

9-0620403 

•001108647 

815409 

736314327 

30-0499584 

9-6050096 

•001107420 

817216 

738763264 

30-0065928 

9-0691762 

•001106195 

819025 

741217025 

30-0832179 

9-6727403 

•001  104972 

82ii836 

743077416 

30-0998339 

9-6763017 

•001103753 

822649 

746142043 

30-1164407 

9-6798604 

•001  102536 

824464 

748613312 

30-1330383 

9-0834166 

001101322 

826281 

751089429 

30-1496209 

9-086'.  (701 

•001100110 

828100 

753571000 

30-1002003 

9-0905211 

•001098901 

829921 

756058031 

30-1827705 

9-6940694 

•001097695 

831744 

758550528 

30-  1993377 

9-0976151 

•001090491 

833569 

761048497 

30-2158899 

9-7011583 

•001095290 

83.13!  >6 

703551944 

302324329 

9-7040!  »H9 

•001094092 

837225 

70006(1875 

30-2489(569 

9-7082369 

•00  1(192896 

839056 

708575296 

30-2654919 

9-7117723 

•001091703 

840889 

771095213 

30-2820079 

9-7153051 

•0010!i()5I3 

842724 

773620U32 

30-2985148 

9-7188354 

•001089325 

SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


177 


Squares. 


Square  R-oots. 


Reciprocals. 


919 

844501 

770151559 

303150128 

9-7223031 

•001088139 

920 

840400 

778088000 

30-3315018 

9-72.18883 

•001086057 

921 

848241 

781220001 

30-3470818 

9-7204109 

•001085776 

922 

850084 

783777448 

30-3644529 

9-7320309 

•001084599 

923 

85  J  929 

78(i330407 

303809151 

9-73(34484 

-  -001083423 

924 

853770 

7888891)24 

30-3073083 

9-7300034 

•001082251 

925 

855025 

791453125 

30-4  13.S  127 

9-7434758 

•001081081 

020 

85747G 

794022776 

30-4302481 

9-7400857 

•001079014 

927 

851)329 

790597983 

30-4400747 

9-7504930 

•001078749 

92^ 

801184 

799178752 

30-4030924 

9-7530979 

•001077586 

921' 

803041 

801765089 

30-4795013 

9-7575002 

•001070426 

930 

804900 

804357000 

30-4959014 

97010001 

•001075209 

931 

800761 

80G954491 

30-5122-)2G 

97644974 

•001074114 

932 

808024 

809557508 

30-5280750 

9-70791)22 

•001072901 

933 

870489 

812100237 

30-5450487 

9-7714845 

•601071811 

934 

872350 

814780504 

30-5014136 

9-7740743 

•001070064 

935 

874225 

817400375 

30-5777097 

9-7784616 

•001069519 

93G 

870096 

820025856 

30-5941171 

9-7819466 

•001068376 

937 

877909 

822050953 

'30-0104557 

9-7854288 

•001067236 

938 

879844 

8252D3G72 

30-G207857 

9-7889087 

•001066098 

939 

881721 

827930019 

30-043]  069 

9-7923801 

•001064903 

940 

883000 

830584000 

30-0594  194 

9-7958611 

•001003830 

941 

885481 

833237021 

30-6757233 

9-7993336 

•001002099 

942 

887364 

835890888 

30-6920185 

9-8028036 

•001061571 

943 

889249 

838501807 

30-7083051 

9-8002711 

•001060445 

944 

891136 

841232384 

30-7245830 

9-8097302 

•001059322 

945 

893025 

843908025 

30-7408523 

9-8131989 

•001058201 

94G 

894916 

840590536 

30-7571130 

9-8100591 

•001057082 

947 

890809 

849278123 

30-7733051 

9-8201109 

•001055966 

948 

898704 

851971392 

30-7890086 

9-8235723 

•001054852 

949 

900001 

854070349 

30-8058436 

9-8270252 

•001053741 

950 

902500 

857375000 

30-8220700 

9-8304757 

•001052032 

951 

904401 

800085351 

30-8382879 

9-8330238 

•001051525 

952 

900304 

862801408 

30-8.544972 

9-8373005 

•001050420 

9515 

908209 

863523177 

30-8700981 

9-8408127 

•001049318 

954 

910116 

808250004 

30-8808904 

9-8442536 

•001048218 

955 

912025 

870983875 

30-9030743 

9-8470920 

•001047120 

956 

913936 

873722816 

30-9192497 

9-8511280 

•001040025 

957 

915849 

870407493 

30-9354166 

9-8545G17 

•001044932 

958 

917704 

879217912 

30-9515751 

9-8579020 

•001043841 

959 

910081 

881974079 

30-9077251 

9-8014218 

•001042753 

900 

921  GOO 

884730000 

30-9838008 

9-8048483 

•001041007 

981 

923521 

887503081 

31-0000000 

9-8682724 

•001040583 

962 

925444 

890277128 

310101248 

9-8716941 

•001039501 

9G3 

9273G9 

893050347 

31-0322413 

9-8751135 

•001038422 

904 

929296 

895841344 

31-0483494 

9-8785305 

•001037344 

905 

931225 

898032125 

31-0644491 

9-8819451 

•001030209 

900 

933156 

901428096 

31-0805405 

9-8853574 

•001035197 

907 

935089 

904231003 

3l-090C)236 

9-8887073 

•001034126 

90S 

937024 

907030232 

31  1120984 

9-8021749 

•001033058 

909 

938961 

909853209 

31-1287(548 

9-8955801 

•001031992 

970 

940900 

912073000 

31  144H230 

9-8080830 

•001030928 

971 

942841 

9  154!  (HO  11 

31  1(508729 

9-91)23835 

•001029866 

972 

944784 

918330048 

31  1769145 

9-9057817 

•001028807 

973 

94(5729 

921107317 

31  1920479 

9-909177(5 

•001027749 

974 

948070 

924010424 

31  -'2080731 

9-9125712 

•00102<)G94 

975 

95(Hi25 

926850375 

3l  2240000 

9-9  159024 

•00  1025*541 

970 

952576 

929714176 

31  2409087 

9-9193513 

•001024590 

977 

954529 

932574833 

31-25G0992 

9-9227379 

•001023541 

978 

95154H4 

935441352 

312720915 

9-92H1222 

•001022495 

979 

958441 

938313739 

31-2889757 

9-9205042 

•001021150 

980 

960400 

941192000 

3]  3040517 

99328839 

001020408 

178 


SQUARES,  CUBES,  ETC.,  OF  NUMBERS. 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

981 

962361 

944076141 

31-3209195 

9-9362613 

001019168 

982 

964324 

946966168 

31-33*58792 

9-931N5363 

•001018330 

983 

966289 

9498621187 

31-3528308 

9-9430092 

•0111017294 

984 

968256 

952763904 

31-3687743 

9-9463797 

•001016260 

985 

970225 

955671625 

31-3847097 

9-9497479 

•001015228 

986 

1)72190 

958585256 

31.4006369 

9-9531138 

•001014199 

987 

974169 

961504803 

31-4165561 

9-956-1775 

•001013171 

988 

976144 

964430272 

31-4324673 

9-9598389 

•001012146 

989 

978121 

967361669 

31-4483704 

9-9*531981 

•001011122 

990 

980100 

970299000 

31-4l542li54 

996(55549 

•001010101 

991 

982081 

973242271 

31-4801525 

9-9699005 

•001009082 

992 

9840(54 

976191488 

31-4960315 

9-9732619 

•001008065 

993 

986049 

9791  4i  5(557 

31-5  119025 

9-9766120 

-0(11007049 

!)94 

988036 

982107784 

31-5277655 

9-979;).~>95) 

•00100(5036 

995 

990025 

985074875 

31-54W206 

9-9833(155 

001005025 

996 

992016 

988047936 

31-55(»4<>77 

9-986«488 

•001004016 

997 

994009 

991026973 

31-5753068 

9'9899'.MM) 

•001003009 

'  998 

i»;  >t  5004 

994011  992 

31-591  J380 

9-9933289 

•001002004 

999 

998001 

997002999 

31-6069613 

9-!>966(;56 

•001001001 

MM) 

1000000 

1000000000 

31-62277(>6 

10-0000000 

•001000000 

1(M)1 

1000201 

1003003001 

3  1  -6:^5840 

10-0033222 

•0009990010 

1002 

1004004 

1006012008 

31-6543S36 

10-0066J582 

•0009980040 

1003 

1006009 

1009027027 

31.6701752 

10-(M)99899 

•0009970090 

1004 

1008016 

1012048IM54 

31-6859590 

]  0-0  133  155 

•0009960159 

JU05 

10J0025 

10  15075  125 

31-7017349 

]()•()  16(5389 

•00(19950249 

1000 

1012036 

1018108216 

31-7175030 

100199601 

•0009940358 

1007 

1014049 

1021147343 

31-7332633 

10-0232791 

•0009930487 

1008 

XI16064 

1024  192.)  12 

31-7490157 

10-0265958 

•00095)20635 

100<J 

1018081 

1027243729 

31-7647603 

10-0299104 

•0009910803 

1010 

1020100 

103030  II  KM) 

31-7804972 

100332228 

•0009900990 

1011 

102i'121 

1033364331 

31-7962262 

10-0365330 

•0009891197 

1012 

1024144 

1036433728 

31-8119474 

10-0398410 

•0009881423 

1013 

1031)169 

1039509197 

31-8276609 

10-0431469 

•0009871668 

1014 

1028196 

1042590744 

31-8433686 

10-0464506 

•00098611133 

1015 

1030225 

1045678375 

31-a590646 

100497521 

•0001)8.12217 

1010 

1032256 

1048772096 

31-8747549 

10-0530514 

•0()0!)842520 

1017 

1034289 

1051871913 

31-8904374 

10-0563485 

Ol)0!)*32842  \ 

1018 

1036324 

1054977832 

31  9061123 

10-059I5435 

0009823183 

1019 

1038361 

lossosgaw 

319217794 

10-0629364 

0009813543 

1020 

1040400 

1061208000 

319374:188 

10-06(58271 

01  «  (9803922 

1021 

1042441 

1064332261 

31-9530906 

10-0695156 

0009794319 

J022 

1044484 

1  067462648 

31-9687347 

10-0728020 

0001*784736 

1023 

1046529 

1070599167 

31-9843712 

10-0760863 

0009775171 

J024 

1048576 

1073741824 

32-0000000 

10-0793684- 

0005)765(525 

1025 

1050625 

1076890625 

320J56212 

]0-082<5484 

000975(5098 

1026 

1052676 

1080045576 

320312348 

10-08592(52 

000974(5589 

1027 

1054729 

108320(5(583 

320468407 

10-0892019 

0009737098 

1(128 

1056784 

1086373952 

320624391 

10-0924755 

0009727626 

1029 

1058841 

1089547389 

32-0780298 

10-0957469 

•0009718173 

1030 

1060900 

1092727000 

320936131 

10-0990163 

•0005)708738 

1031 

1062961 

1095912791 

32  J  09  1887 

10-1022835 

•0009699321 

1032 

1005024 

1095)104768 

321247568 

10-1055487 

•0009689922 

1033 

1067089 

1102302937 

32-  1403  173 

10-1088117 

•6009680542 

1034 

10«>9156 

1  105507304 

32-1558704 

10-1120726 

•0005X57  11  80 

1035 

1071225 

1108717875 

321714159 

10-1153314 

•000966)836 

1031i 

1073296 

1111934656 

32-1869539 

10-1185882 

•0009652510 

1037 

1075369 

11  151  57653 

32-2024844 

10-1218428 

0009(543202 

1038 

1077444 

1  1  I8386H72 

32-2180074 

10-1250953 

0009633911 

1039 

1079521 

1121622319 

32  -23:15229 

10-1283457 

•00096-24  639 

1040 

1081600 

1  134864000 

322490310 

10  -1315941 

•0009015385 

1041 

1083681 

1  1281  1  1921 

32-  21545316 

10-1348403 

•0009606148 

1042 

1085764 

1131366088 

32-2800248 

10-1380845 

•0009596929 

TABLE  XII. 

LOGARITHMS    OF    NUMBERS 
FROM    1    TO    10000. 


TABLE, 


CONTAINING 


THE  LOGARITHMS  OP  NUMBERS 


FROM    1    TO    10,000. 


NUMBERS  FROM  I  TO  100  AND  THEIR  LOGARITHMS, 
WITH   THEIR   INDICES. 


No. 

Logarithm. 

No. 

Logarithm. 

No. 

Logarithm. 

No. 

Logarithm. 

No. 

Logarithm. 

1 

o-oooooo 

21 

1-322219 

41 

1-612784 

61 

1-785330 

81 

1-908485 

2 

0-  301030 

22 

1-342423 

42 

1-623249 

03 

1-792392 

82 

1-913814 

3 

0-477121 

23 

1-361728 

43 

1-033468 

S3 

1-799341 

83 

1-919078 

4 

0-G020GO 

24 

1-380211 

44 

1-643453 

64 

1-800180 

84 

1-924279 

5 

0-0981)70 

25 

1-397940 

45 

1-653213 

65 

1-812913 

85 

1-929419 

6 

0-778151 

26 

1-414973 

46 

1-662758 

66 

1819544 

86 

1-934498 

7 

0-845098 

27 

1-431364 

47 

1-072098 

67 

1-820075 

87 

1-939519 

8 

0-903090 

28 

1-447158 

48 

1-681241 

68 

1-832509 

88 

1-944483 

9 

0-954243 

29 

1-462398 

49 

1-690196 

69 

1-838849 

89 

1-949390 

10 

i-oooaoo 

30 

1-477121 

50 

1-698970 

70 

1-845098 

90 

1-954243 

11 

1-041393 

31 

1-491362 

51 

1-707570 

71 

1-851258 

91 

1-959041 

1-079181 

32 

1-505150 

52 

1-716003 

72 

1-857332 

92 

1-963788 

13 

1-113943 

33 

1-518514 

53 

1-724276 

73 

1-863323 

93 

1-968483 

14 

1146128 

34 

1-531479 

54 

1-732394 

74 

1-F69232 

94 

1-973128 

15 

1-176091 

35 

1-54400H 

55 

1-740363 

75 

1.875061 

95 

1-977724 

16 

1-204120 

36 

1-556303 

56 

1-748188 

76 

1-880814 

96 

1-982271 

17 

1-230449 

37 

1-568202 

57 

1-735875 

77 

1-886491 

97 

1  -1)86772 

18 

1-255273 

38 

1-579784 

58 

1-763428 

78 

1-892095 

98 

1-991226 

19 

1-278754 

39 

1-591065 

59 

1-770852 

79 

1-897627 

99 

1-995635 

20 

1-301030 

40 

1-602060 

60 

1-778151 

80 

1-9U8090 

100 

2-000000 

NOTE. — In  the  following  part  of  the  Table,  the  Indices  are  omitted,  as  *hey 
can  be  very  easily  supplied.  Thus,  the  index  of  the  logarithm  of  every  integer 
number,  consisting  only  of  one  number,  is  0;  of  two  figures,  1;  of  three 
figures,  2;  of  four  figures,  3:  being  always  a  unit  less  than  the  number  of 
figures  contained  in  the  integer  number.  The  index  to  the  logarithm  of  every 
number  above  100,  in  the  following  part  of  the  Table,  is  omitted;  yet.  in  the 
operation,  it  must  be  prefixed,  according  to  this  remark  :  so  that  the  logarithm 
of  600  is  2-77815,  and  that  of  60UO  is  377815,  and  so  of  the  rest. 

180 


LOGARITHMS  OF  NUMBERS. 


181 


Na  |  0  |  I     3  |  3  {  4    5  |  6    7  |  8  |  9  |  Diff. 

100 

(MXKKK) 

000434  1000868100  HOI  1001734  0021(56  002598  (K)3029 

003461 

0038511  1432 

1 

«Bl 

4751 

5181   5(509  j  (5038   646(5   (1894 

7321 

7748 

8  174  '428 

2 

8(500 

902(5 

9451   987(5  10  1030*  i  010724  01  1  147 

011570 

011993 

012415  424 

3 

012837 

013259 

013(180 

014  id!)  i  .  4521   4940   53(10 

5779 

(5197 

6(516  420 

4 

7033 

7451 

78(58 

8284  i  8700   91K5   9532 

9947 

020361 

020775:416 

5 

02  11  HI) 

0-2  1(503 

022016 

022428 

022841  02325-2  (!2.'{()(i4 

024075 

4486 

48961412 

(i 

5308 

5715 

6123 

(5533 

(5942   7350   7757   81(54 

8571 

8!)78  408 

7 

9384 

978S) 

030195 

030(500 

031004  031408  031812  03-2-21(5 

032(519 

03:50-21  404 

8 

0334-24 

033826 

4227 

4628 

5029   5430   58:50 

6230 

6(529 

70281400 

9 

7426 

7825 

8223 

8(520 

91)17   9414   9S11 

040207 

040602 

040998  397 

1JO 

041393 

041787 

042182 

04257(5 

0429(59  0433(52  043755 

044148 

044540 

044932  393 

1 

53-23 

5714 

6105 

6495 

GH85   7275   7(5(5! 

8053 

8442 

8831)  390 

2 

9218 

0606 

9993 

050380 

0507(56  051153  051538  051924  052309 

052(594  386 

3 

053078 

0531(53 

05384(5 

4230 

4613   4996   53781  57601  SJ42 

6524  383 

4 

0905  i  728(5   7066 

8046 

8426   8805   9185   95(53 

9942 

0(50320  379 

5 

060098 

061075 

061452 

061821)  06220(5  062582  062i!o8  063:533 

063709 

4083:376 

6 

4458 

4832 

520(5 

5580 

5953   0326   GG99   7071 

7443 

7815373 

7 

8180 

8557 

81)28 

9298 

9(5(58  070038  070407 

07077(5 

071145 

071514  370 

8 

07  188-2  072-250 

072(517 

072985 

073352 

3718i  4085 

4451 

481(5 

5182366 

9 

5547 

5912 

6276 

6640 

7004 

7368   7731 

8094 

8457 

8819:363 

120 

079  18  J 

079543  079904 

0802(56 

080626  '  080987  1  081347  !  081707 

082067 

082426  360 

1 

082785  083144  083503 

38(51 

4-211)   4576   4934 

5291 

5(547 

(5004  !  357 

2 

63(50   15710 

7071 

7426 

7781   8136   8490 

8845 

9198 

9552  355 

3 

4 

9905  090258 
0934-22   3772 

090(511 
4122 

090963 

4471 

09J315  091(567 
4820]  5169 

092018 
5518 

092:170 
586(5 

092721 
6215 

093071  352 

(15(52  349 

5 

GUI!) 

7257 

7(504 

7951 

8298   8(544 

8990 

9335 

9(581 

100026346 

6 

100371 

100715 

101059 

101403 

10  1747;  102091  102434 

102777 

103119 

3462  343 

7 

3804 

4J4(5 

4487 

4828 

51(59   5510   5851 

6191 

6531 

6871  341 

8 

7210 

7549 

7H88 

8227 

8565   8903   9241 

9571) 

991(5 

110253  338 

9L110SBO 

110926 

111203 

111599 

1119341112270  (  112605 

112940 

113275 

3609  335 

130!  113943  [14-271 

114(511 

114944 

115278  115611  115943 

116276 

11(5608 

116940333 

1 

7271 

7(i()3 

7934 

82(55   85<J5S  892(5   9256 

9586 

9915 

120245  330 

2 

120574 

120903 

121231 

121560  1218H8  J22316  122544 

122871 

123  1  9H 

3525  328 

3 

3852 

4178 

4504 

4830   5156 

5481 

5806 

6131 

6456 

6781  325 

4 

7105 

7429 

7753 

8076   8390 

'8722 

9045 

9368 

9(590 

130012  323 

5 

130334 

130(555 

130977 

131298  1316191  131939  1322(50 

132580 

132900 

3219321 

6 

3539 

3858 

4177 

449(5 

48141  5133 

5451 

5769 

608(5 

6403  318 

7 

(5721 

7037 

7354 

7(571 

7987   8303 

8618 

8934 

9249 

95(54  3J6 

8 

9879 

140194 

140508 

140822 

141136  141450 

1417(53 

142076 

142389 

142702314 

9 

143015 

3327 

3(539 

3951 

4263 

4574 

4885 

5196 

5507 

5818  311 

141) 

140128 

146438 

14(5748 

147058 

147367 

147676 

1  47985  '  14829-1 

148603 

148911  309 

J 

9219 

9527 

9835 

150142 

150449 

15075(5  1510(53  151370 

15167(5 

151982307 

2 

15-2288 

152594 

152900 

3205 

3510 

3815 

4120 

4424 

47-28 

5032  305 

3 

533(5 

5(540 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8001  303 

4 

8362 

8604 

80(55 

9266 

9567 

9868 

1601(58 

1604(59 

160769 

1610(58  301 

5 

161308 

16  J  I5r.7 

MJ1D87 

16226G 

162564 

1(528(5:5 

3161 

34(50 

3758 

4055  299 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

672(5 

7022  297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9074 

.  9968  295 

8 

17021)2 

170555 

170848 

171i4! 

171434 

171726 

172019 

17231  1 

172(503 

172895  293 

9 

318(5 

3478 

3769 

40!  iO 

4351 

4641 

4932 

5222 

5512 

5802291 

150 

17(5091 

176381 

17(5670 

17(5959 

177248 

177536 

177825 

1781  13 

178401 

178689  289 

1   81)77 

92(54 

9552  i  9839 

180  12(5 

180413 

180(599 

18098(5 

181272 

18  J  558  287 

2 

181844 

182129 

182415  182700 

2985 

3270 

3555 

3839 

4123 

4407  285 

3 

41)91 

4975 

5259 

5542 

58-25 

6108 

6391 

6(574 

6956 

7239  283 

4 

75-2  J 

78031  8084 

83(5(5 

8(547 

89-28 

9209 

9490 

9771 

190051  281 

5  190332 

190612!  190892 

191171 

191451 

191730 

192010 

192289 

1925(57 

2846  279 

6 

3125 

3403 

3681 

:i959 

4237 

4514 

4792 

50I5U 

5346 

5(523  278 

7 

59HO 

6176 

(5453 

6729 

7005 

7281 

755(5 

7832 

8107 

8382  276 

8   8657   8932 

9206 

9481 

9755 

200029 

200303  200577  j  200850 

201124  274 

9  20  1397  120  1670  20  1943 

20221(5 

202488 

2761 

3033|  3305   357?|  3848272 

No.   0   |   1     J4  |   3   |  4     5   |   0 

7  |  8  |  0  (Die 

182 


LOGARITHMS  OF  NUMBERS. 


No.  I     O 


3     |     3 


|     5 


9     |D,£ 


160 

204120 

204391 

204663 

204934  20.1204 

20.V475 

205746;206016  20628t> 

206556  271 

1 

6826 

7096 

7365 

76:14 

7904 

8173 

8441   8710 

8979 

9247 

2ii:» 

o 

9515 

9783 

210051 

210319 

210586 

210853 

211121  211388 

211654 

211921 

267 

3 

212188 

212454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

2iK5 

4 

4844 

5109 

5373 

5(538 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

90(50 

9323 

9585 

9846 

262 

6 

220108 

220370 

220631 

220892 

221153 

221414 

221675 

22193(5 

222196 

222456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7(530 

258 

9 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9U82 

9938 

230193 

256 

170 

23J449 

230704 

230960 

231215 

231470 

231724 

231979 

232234 

232488 

232742 

255 

I 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8040 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

240050 

240300 

250 

4 

240549 

240799 

241048 

241297 

24ir>46 

241795 

242044 

242293 

2541 

2790 

249 

5 

3038 

3286 

3534 

3782 

40311 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

600(5 

6252 

(5499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

250176 

245 

8 

250420 

250664 

250908 

251151 

251395 

251638 

251881 

252125 

252368 

2610 

243 

9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

18,) 

255273 

255514 

255755 

255996 

256237 

256477 

256718 

256958 

257198 

257439 

241 

J 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

260310 

260548 

260787 

261025 

261263 

261501 

261739 

261!>76 

262214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

64(57 

6702 

•  6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

8 

9513 

974(5 

9980 

270213 

270446 

270679 

270912 

271144 

271377 

271609 

233 

7 

271842 

272074 

272306 

2538 

2770 

3001 

3233 

34(54 

3696 

3927 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

'.) 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

278982 

27921  1 

279439 

279667 

279895 

280123 

280351 

280578 

280806 

228 

1 

281033 

281261 

281488 

281715 

281942 

282169 

9396 

2622 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

2) 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

95S9 

9812 

223 

5 

290035 

290257 

290480 

290702 

290925 

291147 

291369 

291591 

291813 

292034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

424(5 

221 

7 

4466 

4(587 

4907 

5127 

5347 

5567 

5787 

6007 

622(5 

6446 

220 

8 

66(55 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

y 

8853 

9071 

9289 

9507 

9725 

9943 

300161 

300378 

300595 

300813 

218 

20.) 

301030 

301247 

3014(54 

301681 

301898 

302114 

302331 

302547 

302764 

302980 

217 

1 

3196 

3412 

3(528 

3844 

4059 

4275 

44J1 

4706 

4921 

513li 

216 

2 

5351 

5.-j66 

5781 

5996 

6211 

6425 

6(539 

6854 

7068 

7282  j  2  15 

3 

749*5 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417213 

4 

91)30 

9843 

310056 

3102158 

310481 

31069:5 

31090(5 

311118 

311330 

311542212 

5 

31  1754 

3119(5(5 

2177 

2389 

2(500 

2812 

3023 

3234 

3445 

3656 

211 

8 

3rt67 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

57(50 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

8 

80(53 

8272 

8481 

8689 

88i)8 

911(6 

9314 

9522 

9730 

9938 

208 

y 

32014(5 

320354 

320562 

320769 

320977 

321184 

321391 

321598 

321805 

322012 

207 

210 

32221!! 

32242(5 

322633 

322839 

323046 

323252 

323458 

323665 

323871 

324077 

206 

1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

592(5 

6131 

205 

2 

633(5 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8IJ80 

8583 

8787 

8991 

9194 

9398 

9801 

9805 

330008 

330211 

203 

4 

330414 

330(517 

330819 

331022 

331225 

331427 

331630 

331832 

2034 

$O6 

202 

5 

2438 

2(540 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

9 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

64(50 

66(50 

68(50 

7060 

7260 

7459 

7(559 

7858 

8058 

82.">7 

200 

8 

8456 

84556 

8855 

9054 

9253 

9451 

9650 

9849 

340047 

340246!  199 

9  340444 

340642 

340841 

341039 

341237 

341435 

341632 

341830 

2028 

2225|  198 

No.  1    O     I 


3     I     0 


8  I 


LOGARITHMS  OF  NUMBERS.            183 
No.  |  0  |  1  [  3  |  3  |  4    5  !  6  |  7  i  8  I  9  |  D.« 

220 

342423 

U2020  .'{42817 

343014 

M3212 

34340!) 

343(50(5 

J43802 

343999 

344196 

197 

J 

4392 

458i) 

4785 

4981 

5178 

5374 

5570 

57(50 

5902 

0157 

196 

o 

6353 

0549 

0744 

0939 

7135 

7330 

7525 

7720 

7915 

81  10 

195 

3 

8305 

8500 

8(594 

8889 

9083 

St-278 

9472 

9(5(515 

9*00 

350054 

194 

4 

350248 

350442 

3501530 

J50829 

151023 

351210 

351410 

151603 

351796 

1989 

193 

5 

2183 

2:575 

2508 

2701 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

41  OH 

4301 

4493 

4085 

4870 

50(58 

5-200 

5452 

5043 

5834 

192 

7 

6026 

8217 

(5408 

6599 

6790 

(5981 

7172 

7303 

7554 

7744  191 

8 

7935 

8125 

8316 

8500 

8(596 

8886 

9076 

9-2(5(5 

9451) 

9646 

190 

9 

9835 

360025 

360215 

360404 

360593 

360783 

360972 

301101 

361350 

361539 

189 

530 

3(517-28 

361917 

302105 

302294 

362482 

302(571 

362859 

3G3048 

363536 

363424 

188 

1 

3619 

3800 

3988 

4170 

43(53 

4551 

4739 

4920 

5113 

5301 

188 

2 

5488 

5(575 

5802 

6049 

0230 

0423 

61510 

0790 

0983 

7169 

187 

3 

735(5 

7542 

7729 

7915 

8101 

8287 

8473 

8(559 

8845 

9030 

186 

4 

921(5 

9401 

9587 

9772 

9958 

370143 

370328 

370513 

370(598 

370883 

185 

5 

371008 

371253 

371437 

371(522 

371800 

1991 

2175 

2300 

2544 

2728 

184 

6 

2912 

3090 

3280 

34(54 

3047 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

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5(504 

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6029 

6212 

6394 

183 

8 

6577 

6759 

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8034 

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182 

9 

8398 

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240 

380211 

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381476 

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381837 

181 

1 

2017 

2197 

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2557 

2737 

2917 

3097 

3277 

3456 

303(5 

180 

2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

3 

5(500 

5785 

5904 

0142 

6321 

0499 

0077 

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7034 

7212 

178 

4 

7390 

7508 

7740 

71C3 

8101 

8279 

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8034 

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8989 

178 

5 

9100 

9343 

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9(598 

9875 

390051 

3*0228 

390405 

390582 

390759 

177 

(I 

390935 

391112 

391288 

3914(54 

391041 

1817 

1993 

2109 

2345 

2521 

176 

7 

2097 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

46S7 

4802 

4977 

5152 

532(5 

5501 

5676 

5850 

0025 

175 

9 

0199 

0374 

0548 

6722 

0896 

7071 

7245 

7419 

7592 

7788 

174 

25(1 

397940 

398114 

398287 

398401 

398034 

398808 

398981 

399154 

399328 

399501 

173 

1 

9074 

9847 

400020 

400192 

400305 

400538 

400711 

400883 

401050 

401-228 

173 

2 

401401 

401573 

1745 

1917 

2089 

2201 

2433 

2005 

2777 

2949 

172 

3 

3121 

3292 

3404 

3035 

3807 

3978 

4149 

4320 

4492 

4(563 

171 

4 

4834 

5005 

5176 

5340 

5517 

5088 

5858 

6029 

6199 

6370 

171 

5 

0540 

0710 

0881 

7051 

7221 

7391 

7501 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9420 

9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

4  10(509 

410777 

41094(5 

111114 

411283 

411451 

109 

8 

411020 

1788 

1950 

2124 

2293 

2461 

2029 

2790 

2904 

3132 

108 

<J 

3300 

3467 

3035 

3803 

3970 

4137 

4305 

4472 

4039 

480(5 

167 

261) 

414973 

415140 

415307 

415474 

415041 

415808 

415974 

410141 

410308 

416474 

107 

J 

0041 

0807 

0973 

71X1 

7306 

7472 

7(538 

7804 

7970 

8135 

166 

8301 

8407 

8033 

8798 

8964 

912J 

9295 

94(50 

90-25 

9791 

105 

3 

9950 

420121 

42028(5 

420451 

420(510 

420781 

420945 

4211  JO 

421275 

42J43U 

1(55 

4 

421004 

1708 

1933 

2097 

22(51 

2420 

2590 

2754 

2918 

3082 

1(54 

5 

324(5 

3410 

3574 

3737 

3901 

4005 

4-2-28 

4:59-2 

4555 

4718 

104 

6 

4882 

5045 

5208 

5371 

5531 

5t597 

5800 

6023 

018(5 

0349 

103 

7 

051  1 

607-1 

6830 

0999 

71611  7324 

7480 

7048 

7811 

7973 

102 

8 

8133 

8297 

8459 

8021 

8783   8944 

9100 

9208 

9429 

9591 

102 

9 

9752 

9914 

430075 

430230 

430398  j  430559 

430720 

430881 

431042 

431203 

161 

270 

431304 

43152;) 

431085 

431840 

432007  432107 

432328 

432488 

432049 

432809 

101 

1 

2969 

3130 

329» 

3450 

3(510 

3770 

3930 

4090 

4249 

4409 

1(50 

2 

45(59 

4729 

4888 

5048 

5207 

5307 

5526 

5085 

5844 

6004 

159 

3 

0103 

6322 

6481 

0040 

0799 

6957 

7  1  1(5 

7275 

7433 

7592  159 

4 

7751 

7909 

80(57 

8220 

8381 

8542 

8701 

8859 

9017 

9175 

158 

9:533 

9491 

9048 

980T 

9904 

44012-2 

440279 

440437 

440594 

440752 

158 

C) 

4411909 

441000 

441224 

441381 

441538 

1095 

1852 

2009 

210(5 

2323 

157 

7 

2-18' 

2037 

279; 

2950 

3100 

32(53 

3419 

3576 

3732 

3889 

157 

8 

404i>i  4201 

4357 

4513 

4(509 

4825 

49<-l 

5137 

5293 

5449  156 

91  56041  5760 

5915 

6071 

0220 

6382 

6537 

0(592 

6848 

7003,155 

7|8|9 


184 


LOGARITHMS  OF  NUMBERS. 


Na|0|       1|J8|3|4:|5|G|7|8|9 


2801447158 

4473134474(58 

447623 

447778 

447933 

4480881448-242 

448397 

!4.-'5.V2  I."-.') 

1   8706 

88(51 

9:>15 

9170 

93-24 

9478 

8033   9787 

£41 

4500115 

154 

21  4302-10 

450403 

450557 

45371  1 

450885 

451018 

451172 

45132:; 

451479 

1633 

154 

3 

1788 

1940 

2093 

2-247 

240.) 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3o24 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

.13 

5 

484.1 

4937 

5150 

5302 

5454 

5:506 

575t3 

5iUO 

60G2 

6214 

-,  ) 

6 

6360 

6518 

6(570 

68-21 

6;i73 

7125 

7276 

7428 

7579 

7731 

5-3 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8781) 

8940 

9091 

9242 

51 

8 

93J-2 

9543 

9694 

9845 

9.)1)5 

460146 

46.12.16 

460447 

46051(7 

460748 

151 

S 

4608  J8 

46  1048 

461198 

461348 

461499 

1649 

1799 

1948 

201)8 

2248 

15'J 

290 

4623;!8 

4G2548 

462697 

«52?47 

4C.29J7 

403146 

463296 

463445 

463504 

463744 

150 

3   3893 

4042 

4191 

4340 

4490 

4(539 

4788 

493(5 

5085 

•  5234 

149 

2  5383 

553-2 

51580 

582J 

51)77 

6126 

6274 

(54-23 

6571 

6719 

149 

3 

6868 

701(i 

7164 

7312 

74(50 

7(508 

7756 

7901 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

98-2-2 

9969 

470116 

470'263 

470410 

470557 

470704 

470851 

47091)8 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

3610 

146 

? 

2756 

2903 

3049 

3195 

3341 

3487 

*633 

3779 

3;)-25 

4071 

146 

8 

4216 

4362 

4508 

4(553 

4799 

4944 

5090 

5235 

5381 

552(5 

146 

9 

5671 

5816 

5J62 

6107 

6252 

6397 

6542 

6687 

6832 

697(5 

145 

300 

477121 

47726;; 

477411 

477555 

477700 

477844 

477989 

478133 

478278 

478422 

14.1 

1 

85(56 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

48J2:>4 

480438 

480582 

480725 

480869 

481012 

481156 

4812;)!) 

144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

.2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

G 

57-2 

5863 

6005 

6147 

6281) 

6430 

6579 

6714 

6855 

6997 

142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

82(59 

8410 

141 

8 

8551 

8692 

8833 

8i)74 

9114 

9255 

939(5 

9537 

9677 

98  18 

141 

9 

9958 

490099 

490239 

490380 

4905-20 

490661 

490801 

490941 

491081 

491222 

140 

310 

4913G2 

49150-2 

491642 

491782 

491922 

492062 

41)2201 

492341 

492481  492(521 

140 

1 

27(50 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4-294 

4433 

4572 

4711 

48.10 

4989 

5128 

52(57 

540(5 

139 

3 

5544 

5683 

5822 

5960 

(501)9 

6238 

6376 

6515 

6(553 

6791 

139 

4 

6930 

7068 

7206 

7344 

74H3 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8448 

8586 

87-24 

8862 

899!) 

9137 

9275 

9412 

9550 

138 

6 

9687 

98-24 

99(52 

50001);) 

500-236 

500374 

500511 

500(548 

500785 

500922 

137 

7 

501051) 

501  19S 

501333 

1470 

1607 

1744 

1880 

2017 

2154 

221)1 

137 

8 

24-27 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3(555 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4(507 

4743 

4878 

5014 

136 

320 

5051.50 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

j 

6505 

6640 

6776 

6911 

7046 

7181 

731(5 

7451 

758!i 

7721 

13.1 

2 

7856 

7991 

8126 

82(50 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9-203 

9337 

9471 

UBU6 

9740 

9874 

510009 

510143 

510277 

510411 

134 

4 

510545 

511X579 

510813 

510947 

511081 

511215 

1349 

1482 

1616 

1750 

134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2(584 

2818 

2951 

3084 

133 

6 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

4li81 

4813 

4946 

5079 

5211 

5344 

547(5 

5609 

5741 

133 

8 

5874 

GO!  16 

6139 

6-271 

6403 

6535 

6(5(58 

6800 

6932 

7064 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

3119 

8251 

8382 

132 

330 

5185141518646 

518777 

518909 

519040 

519171 

519303 

519434 

519566 

519697 

131 

JU  98281  9959 

5200.X) 

120221 

520353 

520484 

520615 

520745 

52087(5 

521007 

131 

2 

521138 

521-269 

14  X) 

1530 

1(561 

1792 

1922 

2053 

2183 

2314 

131 

3 

2-444 

2575 

2705 

2835 

29156 

3096 

3226 

3356 

3486 

3(516 

130 

4 

3746 

387(5 

4();)6 

4136 

4266 

4396 

4536 

4656 

4785 

4915 

130 

5 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

65!>8 

6727 

(585'j 

6985 

7114 

7243 

7372 

7501 

129 

7 

7630 

7759 

788S 

8016 

8145 

8274 

8402 

853  1 

86(50 

8788 

129 

8 

8917 

9045 

9174 

930-2 

9430 

9559 

9(587 

9815 

9943 

530072  !  128 

9 

530200 

530328 

530456 

530584 

53071-2 

530840 

530968 

531096 

531223 

1351|  128 

No.  I     O     I     1 


.1     3     |     -1     |     5     | 


J     7 


LOGARITHMS  OF  NUMBERS. 


No.  |     0     |     1 


3      | 


6     |     7 


j     9     |  Dig 


34  J  53l-17i> 

531607  531734 

531802:531990  532117  532245 

532372 

532500 

532627 

128 

1 

2754 

2882 

3009 

313(5   3204 

3391 

3518 

3(545 

3772 

3899 

127 

o 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

J27 

3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

4 

6558,  6(185 

6811 

69'  J7 

7063 

7189 

7315 

7441 

7567 

7693 

126 

5 

7819   7945 

81)71 

8197 

832-2 

8448 

8574 

8699 

8825 

8951 

126 

G 

90761  0-20-2 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

1-25 

540329 

540455 

540580 

540705 

540830 

540955 

541080 

541205 

1330 

1454 

1-25 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

1-25 

9 

9823 

2:)50 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544008 

544192 

544316 

544  MO 

544564 

544688 

544812 

544936 

545060 

545183 

1-24 

1 

5307 

5431 

5555 

5678 

5862 

5925 

6049 

6172 

6296 

6419 

124 

2 

0543 

6666 

6789 

69  1  3 

7036 

7159 

7282 

7405 

7529 

7652 

123 

3 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8-881 

123 

4 

!K)()3 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

550106 

123 

5 

550228 

550351 

550473 

550595 

550717 

550840 

550962 

551084 

551206 

1328 

122 

C 

1450 

157-2 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

7 

26G8 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4(510 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

3GO 

556303 

556423 

556544 

556664 

556785 

556905 

557026 

557146 

557267 

557387 

120 

] 

7507 

7027 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

1-20 

2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

3 

9907 

5600-26 

560  140 

560265 

560385 

560504 

560(524 

560743 

560803 

5(50982 

119 

4 

561J01 

1221   1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

5 

2-293 

24121  2531 

2C-50 

2769 

2887 

-  3006 

3125 

3244 

3362 

119 

6 

3481 

3600   3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4660 

4784 

•  4903 

502  1 

5139 

5257 

5376 

5494 

5612 

5730 

118 

8 

5848 

5966 

6084 

6-21  !-2 

(53-20 

6437 

6555 

6073 

6791 

6909 

118 

9 

7020 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

568202 

568319 

568436 

568554 

568671 

5G8788 

568905 

569023 

569140 

569257 

117 

] 

93:4 

9491 

9I508 

9725 

9842 

9959 

57007(5 

570193 

570309 

570426 

117 

2 

570543 

570660 

570776 

570893 

571010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2872 

2988 

3104 

3220 

333(5 

3452 

3568 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4(510 

4726 

4841 

4957 

5072 

116 

6 

5188 

5303 

5419 

5.134 

5650 

5765 

5880 

5996 

6111 

(5-226 

115 

7 

6341 

6457 

6572 

6G87 

6802 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7007 

7722 

78315 

7951 

8000 

8181 

8-295 

8410 

8525 

115 

9 

8639 

8754 

8868 

8983 

9097 

9212 

932(5 

9441 

9555 

9669 

114 

380 

570784 

579898 

580012 

580120 

580241 

580355 

580469 

580583 

580697 

580811 

114 

1 

580925 

581039 

1153 

12(57 

1381 

1495 

1608 

1722 

1836 

1950 

114 

2 

3063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

3 

3199 

3312 

3426 

3539 

3(552 

3765 

3879 

3992 

4105 

4218 

113 

4 

4331 

4444 

4557 

4670 

4783 

489(5 

5009 

5122 

5235 

5348 

113 

5 

5461 

5574 

51)86 

5799 

5912 

6024 

6137 

6250 

6362 

(5475 

113 

6 

6587 

6700 

6812 

6925 

7037 

714J 

7262 

7374 

7486 

7599 

112 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

ci 

8833 

8944 

905(5 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112 

9 

9950 

590061 

590173 

590284 

590396 

590507 

59061'J 

590730 

590842 

590953 

112 

390 

591065 

591176 

591287 

591399 

591510 

591621 

:91732 

591843 

591  955  '•  592066 

111 

J 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

0 

3-286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282,111 

3 

4393 

4503 

4(514 

4724 

4834 

4945 

5055 

5165 

5276 

5386  110 

4 

5496 

5606 

57-17 

5827 

5937 

(5047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707   6817 

6927 

7037 

7146 

725(5 

73(56 

74761  758(5110 

6 

7(595 

7H05   7914 

8024 

8134 

8243 

8353 

84(52 

8572 

868L  1  10 

7 

8791 

89001  9009 

91  19   9228 

9337 

9446 

955(5 

9665 

9774  109 

8 

9883 

9992161)0101 

600210 

(00319 

600428  (500537 

600(546 

600755  61)0864  H'9 

9 

60097  3  |  60  1082  j  1191 

1299 

141,8   1517J  1625|  17:54 

18431  1951  109 

No.  |      O       | 


I     7     I     8 


186 


S  OF  NUMBERS. 


No.|0|l|3|3|4-|5    G    7  |  8  |  9  TDK 

400 

6C20i50  0021(59 

G02277 

802386 

002494 

602603  002711 

602819 

602928 

60303!! 

108 

1 

3144 

3253 

3361 

34(59 

3577 

3G8G 

3794 

3932 

4010 

4118 

108 

2 

4226 

4334 

4442 

4550 

4(558 

47G6 

4874 

4982 

5089 

5197 

108 

3 

5305 

5413 

5521 

5628 

573(5 

5844 

5951 

6059 

GJGG 

6274 

103 

4 

6381 

6489 

6590 

6704 

6811 

6919 

7026 

7133 

7211 

7348 

107 

5 

7455 

7562 

7G69 

7777 

7884 

7991 

8098 

8-205 

8312 

8419 

107 

6 

85261  8G33 

8740 

8847 

8954 

9061 

91G7 

9274 

93dl 

9488 

107 

7 

9594!  9701 

9808 

9914 

610021 

610128 

610-234 

610341 

610447 

610554 

107 

8 

010060  G107G7 

610873 

G10979 

108G 

1192 

1298 

1-105 

1511 

1G17 

106 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 
1 

G12784 

3842 

612890 
3947 

612996 
4053 

613102  613207 
4159   4264 

6133131613419 
4370   4475 

613525 
4581 

G13G30 
4686 

613736 
4792 

106 

106 

0 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5(534 

574!) 

5845 

105 

3 

5950 

6055 

6160 

6265 

6370 

G47G 

6581 

CG86 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

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7525 

76-29 

7734 

7839 

7943 

105 

5 

8048 

8153 

8257 

8362!  8460 

8571 

8G76 

8780 

8884 

8989 

105 

6 

9093 

9198 

9302 

9406   9511 

9615 

9719 

9824 

9928 

620032 

104 

•J1 

62013(5 

620240 

620344 

620448 

623552 

620656  620700 

6208G4 

6209G8 

1072 

104 

8 

1176 

1280 

138-1 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

9 

2214 

2318 

2421 

2525 

2628 

.  2732 

2835 

2939 

3042 

3146 

104 

420 

6232-19 

623353 

623456 

623559 

623663 

623766 

623869 

G23973 

624070 

624179 

103 

1 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6133 

6-238 

103 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

716! 

72(53 

103 

4 

73(56 

74C8 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

5 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

920G 

9308 

102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

630921 

630123 

630224 

G3032G 

102 

7 

630428 

630530 

630(531 

630733 

630835 

630936 

1038 

1139 

1241 

1I54-2 

102 

8 

1444 

1545 

1(547 

1748 

1849 

1951 

2352 

2153 

2255 

2356 

101 

9 

2457 

2559 

2GGO 

2701 

2862 

2903 

3004 

3165 

32(56 

3367 

101 

430 

633468 

633569 

G33G70 

633771 

633872 

633973 

634074 

634175 

634276 

G34376 

101 

1 

4477 

4578 

4679 

4779 

4883 

4981 

5081 

5182 

5283 

5383 

101 

2 

5434 

5584 

5685 

5785 

5886 

598(5 

6087 

6187 

6287 

6388 

100 

3 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

100 

4 

7490 

7590 

76UO 

7791) 

7893 

7990 

8090 

8190 

8290 

8389 

103 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

100 

C 

9486 

9586 

9686 

9785 

9885 

9984 

G4(K)84 

640183 

640283 

(540382 

99 

640481 

640581 

(540680 

640779 

640879 

640978 

1077 

1177 

127(5 

1375 

93 

8 

1474 

1573 

1G72 

1771 

1871 

1970 

21)69 

2168 

2267 

2366 

99 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

93 

440 

643453 

643551 

643650 

643749 

643847 

643946 

644044 

644143 

614242 

G44340 

98 

1 

4439 

4537 

4(536 

4734 

4832 

4931 

5029 

5127 

522!i 

53-24 

98 

c 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

(5306 

95 

3 

6404 

6532 

6600 

6698 

679(5 

6894 

6992 

7089 

7187 

7283 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872 

7989 

81)67 

81(55 

82(52 

98 

« 

8360 

8458 

8555 

8053 

8750 

8848 

8945 

9043 

9140 

9237 

97 

( 

9335 

9432 

9530 

9327 

9724 

9821 

91)19 

6500  1C 

650113 

97 

7 

650308 

(550405 

653502 

650599 

650(59(5 

650793 

050890 

0987 

1084 

'  1181 

97 

8 

1278 

1375 

1472 

1569 

1(566 

17(52 

185< 

195( 

2053 

2150 

97 

i 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

450 

653213 

653309 

653405 

653502 

653598 

G53G95 

G53791 

653888 

653984 

654080 

90 

1 

4177 

4273 

4361 

4465 

45(52 

4658 

4754 

4851 

4946 

5042 

96 

c 

5133 

5235 

5331 

5427 

5523 

5(519 

5715 

58K 

59!  H 

8003 

96 

5 

60D8 

(5194 

G290 

G38G 

6482 

6577 

6673 

6769 

6864 

6960 

96 

i 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

77°5 

782 

791C 

96 

i 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

( 

8965 

9001 

9155 

9250 

9346 

9441 

9536 

9G31 

972C 

9821 

95 

'  9:uo 

660011 

(5G010< 

660201 

66()2<)G 

660391 

660486 

660581 

660(57* 

GG0771 

95 

8 

G60865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

17J8I  95 

( 

1813 

1907 

2002 

2U6 

2191 

226 

2ItO 

2475 

2)<9 

2663  1  95 

No.   0   |  1  |  »  |  3  1  4-  |  5     fi  "l  7  I  *   |  «   I.,IE 

LOGARITHMS  OF  NUMBERS. 


187 


No.  |     0 


1      I     3     [ 


|      7     I      8      |     9      |D.ff. 


400  662758  ,  C62852,  662947  663041  (563135  663230 
1|  37011  3795   3889  1  3983   4078   417-3 

563324  663418 
42(56   4360 

863512  663(507)  94 
4454   4548!  94 

2   48421  4736   4830!  49-24 

-  5018 

5112 

5206   5299 

5393   5487|  94 

3 

5581 

5675 

5769 

58(52 

5956 

6050 

(5143   6237 

6331!  6424  94 

4 

65  J  8 

6612 

6705 

6799 

6892 

698(5 

7079   7173 

7-2f>»i   7360'  94 

5 

7453 

7546 

7640 

7733 

7886 

71*30 

8013 

8106 

8199   8293 

93 

6 

838(5 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131   9224 

93 

7 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

670060  670153 

93 

8 
9 

670-246 
1173 

570339 
1265 

670431 
1358 

670.124 
1451 

70617 
1543 

670710 
1636 

670802  670895 
1728   1821 

0988 
1913 

1080 
2005 

93 
93 

470 

372098 

672190 

672283 

672375 

672467 

672560 

672652672744 

672836 

672929 

92 

1 

30-21 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

92 

2 

394-2 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

4 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

5 

G694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

680063 

680154 

680245 

91 

9 

680336 

680426 

680517 

680607 

680698 

680789 

680879   0970 

1060 

1151 

91 

480 

681241 

681332 

681422 

681513 

681603 

681693 

681784 

681874 

681964 

682055 

90 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

2 

3047 

3137 

3227 

33  J  7 

3407 

3497 

3587 

3(577 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

4 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

5 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

690019 

690107 

89 

490 

690196 

690285 

690373 

6904(52 

690550 

690639 

690728 

690816 

690905 

690993 

89 

1 

1081 

1170 

1258 

1347 

14:55 

1524 

1612 

1700 

1789 

1877 

88 

2 

1905 

2053 

2142 

2230 

2318 

240(5 

2494 

2583 

2(571 

2759 

88 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903   3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

5 

4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

62(59 

87 

7 

6356 

6444 

6531 

6618 

6706 

(5793 

6880 

6968 

7055 

7142 

87 

8 

722U 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8101 

8188 

8275 

8302 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

698970 

699057 

699144 

699231 

699317 

699404 

699491 

699578 

699664 

699751 

87 

] 

9838 

9924 

700011 

70009& 

700184 

70027  1 

700358 

700444 

700531 

700617 

87 

2 

700704 

700790 

0877 

0963 

1050 

1136 

1-2-22 

1309 

1395 

1482 

86 

2 

1568 

11)54 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

4 

£431 

2517 

2603 

2689 

O**7j 

2861 

2947 

3033 

3119 

3205 

86 

5 

'J291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5(507 

5693 

5778 

86 

8 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6153'. 

85 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

707655 

707740 

70782* 

707911 

707996 

708081 

708166 

708251 

708336 

85 

J 

8421 

8500 

8591 

867( 

8761 

8846 

8931 

9015 

9100 

9185 

85 

S 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

710033 

85 

3 

710117 

7102th. 

710287 

710371 

710456 

710540 

710625 

7107JO 

710794 

0879 

85 

4 

096: 

1048 

1132 

121" 

1301 

1385 

1470 

1554 

1639 

1723 

84 

5 

1807 

189V. 

1976 

2060 

2144 

2221 

2313 

2397 

2481 

2566 

84 

6 

2650 

27:54 

2818 

290- 

2986 

3070 

3154  j  3238 

332; 

340" 

84 

7 

3491 

3575 

3(559 

3742 

382» 

3910 

39941  40781  4JC.2 

4iJ4r 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

84 

9j  5167 

5251 

•5335 

5418 

55U2|  5586 

5669 

5753 

5836 

5920 

84 

No. 


6     | 


|     8 


9     |  Diff 


188 


LOGARITHMS  OF  NUMBERS. 


No.  |     0 


3     f     3 


|     5     |     6     |     7     |     8     f     9     fniit 


520 

716003 

716087 

710170 

716254 

716337 

716421 

716504 

716588 

71(5(571 

71(5 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7:504 

o 

7671 

7754 

7837 

7920 

8003 

808(5 

8169 

•8253 

8336 

8- 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

91(55 

9 

4 

9331 

9414 

9497 

9580 

96(53 

9745 

9828 

9911 

9994 

7-20 

5 

720159 

720242 

720325 

720407 

720490 

720573 

720(555 

720738 

72082  11  0 

6 

098(5 

1068 

1151 

1-233 

1310 

1398 

1481 

15(53 

164(5 

1 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2409 

t> 

8 

2)534 

2716 

2798 

2881 

29(53 

3045 

3127 

3209 

3291 

3 

9 

3450 

3538 

3620 

3702 

3784 

3860 

3948 

4030 

4112 

4 

530 

724270 

724358 

724440 

724522 

724604 

724685 

724767 

724849 

72493  1 

725 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5(5(57 

5748 

5 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

(55(54 

li 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7. 

4 

7541 

7(523 

7704 

7785 

786(5 

7948 

802S 

8110 

8191 

8- 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9 

6 

91(55 

924(5 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9 

7 

9974 

730055 

730136 

730217 

73D298 

730378 

730459 

730540 

730(521 

730 

8 

730782 

0863 

0944 

1024 

1105 

11815 

126(5 

1347 

1428 

1 

9 

158J 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

o 

540 

732394 

732474 

732555 

732635 

732715 

732796 

73287(5 

732956 

733037 

733 

1 

3197 

3273 

3358 

3438 

3518 

3598 

3679 

375!  t 

3839 

3( 

2 

3993 

4079 

4160 

4240 

4320 

4400 

4480 

45!50 

4(540 

4' 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

4 

5593 

5679 

5759 

5^3J 

5918 

5998 

(5078 

6157 

0237 

o: 

5 

6397 

6476 

655(5 

0(535 

6715 

(5795 

0874 

(5954 

7034 

7 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7(570 

7749 

7829 

75 

7 

7987 

8067 

8140 

8225 

8305 

8384 

8463 

8543 

8022 

8' 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9250 

9335 

9414 

9 

9 

9572 

9651 

9731 

9810 

9889 

9968 

740047 

740120 

740205 

740- 

550 

740363 

740442 

740521 

740600" 

740678 

740757 

740830 

740915 

740994 

74  1  ( 

I 

1152 

1230 

1309 

13K8 

1467 

1540 

1024 

1703 

1782 

u 

2 

1939 

2018 

2090 

2175 

2254 

2332 

2411 

2489 

25(58 

21 

3 

2725 

2804 

2882 

29(51 

3039 

3118 

3190 

3275 

3353 

3- 

4 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

40.58 

4130 

4- 

5 

4293 

4371 

4449 

4528 

46015 

4(584 

4702 

4840 

4919 

4' 

6 

5075 

5153 

5231 

5309 

5387 

54(55 

5543 

5021 

5699 

5' 

7 

5855 

5933 

6011 

0089 

6167 

6245 

(5323 

0401 

0479 

(>. 

8 

(5(534 

6712 

6790 

0808 

6945 

7023 

7101 

7179 

725(5 

7; 

9 

7412 

7489 

7567 

7(545 

7722 

7800 

7878 

7955 

8033 

* 

560 

748188 

74826(5 

748343 

748421 

748498 

748570 

748(553 

748731 

748808 

748? 

] 

8963 

9040 

9118 

9195 

9272 

9350 

9127 

9504 

9.582 

9' 

2 

973(5 

9814 

9891 

99(58 

750045 

750123 

750200 

750277 

750354 

750 

3 

751)508 

75058(5 

750(5(53 

750740 

0817 

0894 

0971 

1048 

1125 

1- 

4 

1279 

1350 

1433 

1510 

1587 

1004 

1741 

1818 

1895 

1' 

5 

2048 

2125 

220-2 

2279 

23511 

2433 

2509 

2580 

2663 

2 

6 

281(5 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3. 

7 

3583 

3660 

3736 

3813 

3889 

3900 

4042 

4119 

4195 

4 

8 

434^ 

4425 

4501 

4578 

4(554 

4730 

4807 

4883 

4960 

5 

9 

5112 

5189 

5205 

5341 

5417 

5494 

5570 

5040 

5722 

5 

570 

755875 

755951 

750027 

750103 

756180 

750250 

750332 

756408 

756484 

750 

I 

6030 

6712 

6788 

08154 

6940 

7016 

7092 

7108 

7244 

7. 

2 

7390 

7472 

7548 

7(524 

7700 

7775 

7851 

7927 

8003 

8 

3 

8155 

8230 

8300 

8382 

8458 

8533 

8009 

8085 

8761 

8f 

4 

8912 

8988 

9003 

9139 

9214 

9290 

930(5 

9441 

9517 

9 

5 

90(58 

9743 

9819 

9894 

9970 

760045 

70.)  121 

76019(5 

760272 

700: 

6 

760422 

760498 

760573 

7001149 

700724 

0799 

0875 

0950 

1025 

1 

7 

1170 

1251 

1321) 

1402 

1477 

1552 

1(527 

1702 

1778 

If 

8 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

21 

9 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

3203 

3278 

3: 

No.        0      | 


|     J4 


8          9 


LOGARITHMS  OF  NUMBERS. 


189 


Nu.  |  0  1'  .  1  |  «  |  3    4  |  5  |  6    7  |  8  |  9  |  Diff. 

580 

763428 

763503 

7(53578 

763(553 

763727]  763802 

763877 

763952 

764027 

764101 

75 

1 

417(5 

4251 

4326 

4400 

4475 

4550 

4(124 

4699 

4774 

4848 

75 

o 

4923 

495W 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

3 

5069 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

4 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

5 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

G 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

849(1 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

770042 

74 

9 

770115 

770189 

770263 

770336 

770410 

770484 

770557 

770631 

770705 

0778 

74 

590 

770852 

770926 

770999 

771073 

771146 

771220 

771293 

771367 

771440 

771514 

74 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

2 

2322 

23!)5 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

''4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

6 

524(5 

5319 

5392 

5405 

5538 

5610 

5683 

5756 

5829 

5902 

73 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354  73 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079,  72 

600 

778151 

778224 

778296 

778368 

778441 

778513 

778585 

778658 

778730 

778802  72 

I 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

2 

9590 

9(569 

9741 

9813 

9885 

9957 

780029 

780101 

780173 

780245 

72 

3  780317 

780389 

780461 

780533 

780605 

780677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

197] 

2042 

2114 

2186 

2258 

2329 

2401 

72 

C 

2473 

2544 

2(516 

2688 

.  2759 

2831 

2902 

2974 

304(5 

3117 

72 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

785401 

785472 

785543 

785615 

785686 

785757 

783828 

785899 

785970 

71 

1 

6041 

6112 

6183 

62.54 

6325 

6396 

64G7 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8(363 

8734 

8804 

71 

5 

8875 

8946 

9016 

9:;R7 

9157 

9228 

9299 

93(59 

9440 

9510 

71 

6 

9581 

9051 

9722 

9792 

9863 

9933 

790004 

790074 

790144 

790215 

70 

7 

7902«5 

790356 

790426 

79049G 

790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

9 

JC91 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602 

792672 

792742 

792812 

792882 

792952 

793022 

70 

1 

3092 

3102 

3231 

3301 

3371 

3441 

351  1 

3581 

3651 

3721 

70 

2 

3799 

3860 

3930 

4000 

4070 

4139 

42.19 

4279 

4349 

4418 

70 

3 

4488 

4558 

4627 

4697 

4767 

4836 

49t)!5 

4976 

5045 

5115 

70 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5(572 

5741 

5811 

70 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

643(5 

6505 

69 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

7 

72158 

7337 

7406 

7475 

7545 

7614 

7C83 

7752 

7S2J 

7890 

69 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 
1 

791)341 
300029 

799409 

800098 

799478 
800167 

799547 
800236 

799616 
800305 

799685 
30037* 

799754 

800442 

799823 
800511 

799892 

800580 

799961 

800(548 

69 

69 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1123 

1198 

1266 

1335 

69 

3 

1404 

1172 

1541 

1609 

1678 

1747 

1H15 

18H4 

1952 

2D21 

69 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

25(58 

2637 

2705 

68 

5 

2774 

28-12 

2910 

2979 

3047 

3116 

3184 

32.V2 

3321 

3389 

68 

C 

3457 

3525 

3594 

36R2 

3730 

3798 

3bfi7 

3935 

4003 

4071 

68 

7 

4139 

4208 

4276 

4344 

4412 

4480 

45-18 

4616 

4685 

4753 

68 

8 

4821 

4889 

4957 

5(125 

5093 

5161  !  52.9 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

57731  5841|  5908 

5970 

(1044 

6112 

68 

No.  |     0     |     1 


j     3     | 


I     5     I     6     |     7 


9       I   Drff. 


190 


LOGARITHMS  OF  NUMBERS. 


0     I     1 


|     5     |     6 


8     |     9 


640;  800  180  806218  8063*6  1  306384]  806451]  80651  9  800587 

80(5655  806*23 

S00790  08 

j 

6858 

6926 

6994   706l|  712! 

7197 

72(54 

7332   7400 

7467;  68 

t 

7535 

7003 

7<5?( 

7738!  78()( 

7873 

7941 

8008 

807(5 

8143  68 

J 

8211 

8279 

834f 

P414 

8481 

854! 

861(5 

8t;*4 

875! 

8818 

67 

4 

8886 

8953 

9021 

9088 

9l5f 

9223 

9290 

9338 

9425 

9492 

67 

j 

9560 

9(527 

9(194 

9762 

9829 

9896 

9904 

810031 

810098 

810165 

67 

6 

8  J  0233 

810300 

8103(57 

810434 

810501 

81050! 

810(i3t 

0703 

0770 

0837 

(57 

7 

0904 

0971 

J039 

1101 

1173 

1240 

1307 

1374 

1441 

1508 

(57 

8 

1575 

1042 

1709 

1776 

1843 

19'0 

1977 

2044 

2111 

2178 

67 

9 

2245 

2312 

2379 

2445 

2512 

2579 

264f 

2713 

2780 

2847 

67 

650 

812913 

812980 

813047 

813114 

813181 

813247 

813314 

813381 

813448 

813514 

67 

J 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

(57 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

(57 

2 

4913 

4980 

5046 

5113 

5179 

5246 

531- 

5378 

5445 

5511 

66 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6173 

66 

3 

'  6241 

6308 

6374 

6440 

6506 

6573 

6631 

6705 

6771 

6838 

66 

6 

G'J()4 

0970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

7505 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

8 

8236 

8292 

8358 

8424 

8490 

8556 

81522 

8688 

8754 

882( 

66 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

819610 

819676 

819741 

819807 

819873 

819939 

820004 

820070 

8201  3f 

66 

1 

820-201 

820267 

820333 

t>2«399 

820464 

820530 

820595 

0061 

0727 

0:92 

66 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

C3 

4 

2168 

2233 

2-299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3805 

3930 

3998 

4001 

65 

7 

4126 

4191 

425G 

4321 

438(5 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

516(5 

5231 

5296 

5361 

65 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

65 

670 

826075 

826140 

826204 

826269 

826334 

826399 

82(5464 

826528 

826593 

826658 

65 

1 

6723 

6787 

6852 

6917 

6981 

70-16 

7111 

7175 

7240 

7305 

65 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8-102 

8467 

8531 

8595 

64 

4 

8060 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

5 

9304 

9368 

6432 

9497 

9561 

91525 

9li!K) 

9754 

9818 

9882 

64 

« 

9947 

83001  1 

830075 

830139 

830204 

830268 

830332 

830396 

830460 

830525 

64 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

8 

1230 

1294 

1358 

1422 

I486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

832573 

832637 

832700 

832704 

832828 

832892 

832956 

833020 

833083 

64 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

64 

2 

3784   3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

3 

4421 

4484 

4548 

4011 

40,75 

4739 

4802 

4866 

4929 

4993 

64 

4 

5050 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

55(5-1 

5(527 

63 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

6 

6324 

6337 

6451 

6514 

6577 

6(541 

6704 

6767 

6830 

6894 

63 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

8 

7388 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

82J9 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

(53 

690 

838849 

838912 

838975 

839038 

839101 

839164 

83922? 

839289 

839352 

839415 

63 

1 

9478 

9541 

9(504 

96(5? 

9729 

9792 

9855 

9918 

9981 

840043 

63 

2 

8401  )6 

840169 

840232 

840294 

840357 

840420 

840482 

8411545 

S401508 

0(571 

63 

3 

0733 

079(5 

0859 

0021 

0984 

104(5 

1109 

1172 

1234 

121)7 

63 

4 

1359 

1422 

14H5 

1547 

1610 

1(572 

1735 

1797 

18(50 

1922 

63 

5 

1985 

2047 

2110 

2172 

2235 

2297 

23(50 

2422 

2484 

2517 

(52 

6 

2609 

2072 

2734 

2796 

2859 

2921 

29H3 

3046 

3108 

3170 

(52 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3(50(5 

3869 

3731 

3793 

(52 

8 

3855   3!)  18 

39^0 

4042 

4104 

41(56 

4229 

4291 

4353 

4415 

(52 

9 

4477!  4539 

4601 

46154 

4726 

4788 

4850 

4912 

4974 

5036 

62 

No,  |     0 


3     I     4:     J     5 


D>tf. 


LOGARITHMS  OF  NUMBERS. 


191 


No.  |     0      |     1 


3      I      4 


6     |     7     |     8     |     9     |Diff. 


700 

84509R  845160 

845222 

845284 

845346 

845408 

84547 

845532  845594 

84.5656 

62 

57J8 

5780 

5842 

5904 

5966 

6028 

'  6090 

6151 

6213 

6275 

62 

E 

6337 

6399 

6461 

6523 

6585 

6640 

6708 

6770 

6832 

6894 

62 

j 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

^ 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

« 

8189 

8251 

8312 

8374 

8435 

849" 

8559 

8620 

8(582 

8743 

62 

( 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

93.58 

61 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

8 

85003:5 

850095 

R50156 

850217 

850279 

850340 

85040 

8504C.-2 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

851320 

851381 

851442 

851503 

R51564 

851625 

851686 

851747 

851809 

61 

1 

1870 

1931 

19!t2 

203 

2114 

2175 

2236 

2297 

2358 

2419 

61 

2 

2480 

2541 

2602 

2(563 

2724 

27R5 

2H4f 

2907 

2968 

3029 

61 

2 

3090 

3150 

3211 

3272 

3333 

3394 

345. 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4i&5 

4245 

61 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

85745' 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

1 

7935 

7995 

8056 

8U( 

8176 

823f 

8297 

8357 

8417 

8477 

60 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9K79 

60 

4 

9739 

9799 

9859 

95*  lh 

9978 

860038 

860098 

860158 

860218 

860278 

60 

5 

860338 

860398 

P60458 

860518 

860578 

0637 

0697 

0757 

0817 

0877 

60 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

1534 

1594 

16.54 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

y 

2728 

2787 

2847 

2906 

29B6 

3025 

3085 

3144 

3204 

3263 

60 

730 

863323 

863382 

(753442 

863501 

863561 

863620 

863680 

863739 

863799 

863858 

59 

J 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

2 

4511 

4570 

4630 

4fi«) 

4748 

4808 

486" 

492( 

4985 

5045 

59 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

5 

G287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

6 

6878 

6937 

6996 

7055 

7114 

7173 

723L 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7K*0 

7831 

7998 

59 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

59 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59 

1 

9818 

9877 

9935 

9994 

870053 

870111 

870170 

870228 

870287 

870345 

59 

370404 

870462 

870521 

870579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

14.56 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

6 

2739 

2797 

2855 

£913 

2972 

3030 

3088 

3146 

3204 

32(52 

58 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

8 

3902 

39*50 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

875061 

875119 

875177 

875235 

875293 

875351 

875409 

875466 

875524 

875582 

58 

J 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

2 

6218 

6276 

6333 

6391 

0449 

6507 

6564 

6622 

6(580 

6737 

58 

3 

6795 

6853 

6'JK) 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

6 

8522 

8579 

8637 

86JM 

8752 

8809 

8866 

8924 

8981 

9039 

57 

7 

9096 

9153 

9211 

9268 

9!J25 

9383 

9440 

9497 

9555 

9612 

57 

8 

9069 

9726 

9784 

9841 

9898 

9950 

880013 

880070 

#0127 

3801KJ 

57 

9880-.M2 

88C299 

880356 

880413 

880471 

880528 

0585 

0642 

0699   0756  i  57 

fc. I  o 


192 


LOGARITHMS  OF  NUMBERS. 


No  I     0     I      1     I     3     I     3     I     4:     I     5     |     G     |     7     |     8 


9     |DiS 


760 

880814 

880871 

880.  >28 

880985 

881042  881099 

881150 

881213  881271 

881328 

57 

1 

1385 

1442 

149!) 

155*6 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2120 

2183 

2240 

2297 

2354 

2411 

24(58 

57 

2 

2525 

2581 

2(538 

2(595 

2752 

2809 

28(5(5 

2923 

2080 

3037 

57 

4 

3093 

3150 

3207 

3-2(54 

3321 

3377 

3434 

3491 

3548 

3605 

57 

*j 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

40.19 

4115 

4172 

57 

6 

422  » 

4285 

4342 

4399 

4455 

4512 

45(59 

4625 

4(582 

4739 

57 

7 

4?!).-> 

4852 

4909 

4905 

5022 

5078 

5135 

5192 

5248 

5305 

57 

8 

5301 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

9 

5926 

5983 

6039 

609(5 

6152 

6209 

6265 

6321 

(5378 

6434 

56 

770 

886491 

886547 

886004 

880(5(50 

886716 

886773 

886829 

886885 

880942 

886998 

50 

1 

7054 

7111 

7167 

72'VJ 

7880 

733(5   7392 

7449 

7505 

7561 

56 

2 

7617 

7674 

773i  ) 

7780 

7842 

7898 

7955 

8011 

8067 

8123 

50 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8510 

8573 

8629 

86ar> 

515 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

924(5 

56 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9(538 

9094 

9750 

9800 

56 

6 

98G2 

9918 

9974 

890030 

890086 

890141 

890197 

890253 

890309 

890305 

56 

7 

890421 

890477 

890533 

0589 

9(545 

0700 

0756 

0812 

08(58 

0924 

56 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

9 

1537 

1593 

1649 

1705 

1760 

18115 

1872 

1928 

1983 

2039 

56 

780 

892095 

892150 

892206 

892202 

892317 

892373 

892429 

892484 

892540 

892595 

56 

1 

2051 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

309(5 

3151 

56 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4048 

4704 

4759 

4814 

55 

5 

4870 

4925 

4980 

503(5 

5091 

514(5 

5201 

5257 

5312 

5367 

55 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5804 

5920 

55 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6300 

6301 

6416 

6471 

55 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6907 

7022 

55 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897627 

897682 

897737 

897792 

897847 

897902 

897957 

898012 

898007 

898122 

55 

1 

81  76 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8(515 

8(570 

55 

2 

8725 

8780 

8835 

&890 

8944 

8999 

9054 

9109 

91(54 

9218 

55 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

971  1   9766 

55 

4 

9821 

9875 

9930 

9985 

90003:) 

900094 

900149 

900203 

900258 

900312 

55 

5 

90031)7 

900422 

900476 

900531 

0580 

0040 

0695 

0749 

0804 

0859 

55 

6 

0913 

09(38 

1022 

1077 

1131 

1180 

1240 

1295 

1349 

1404 

55 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

8 

2003 

2057 

2112 

2160 

2221 

2275 

2329 

2384 

2438 

'  2492 

54 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

903090 

903144 

903199 

903253 

903307 

903301 

903416 

903470 

903524 

903578 

54 

1 

3633 

3687 

3741 

3795 

3849 

3904 

39.58 

4012 

4060 

4120 

54 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4001 

54 

3 

4716 

4770 

4824 

4878 

4932 

498(5 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

552(5 

5580 

5634 

5688 

5742 

54 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

6 

6335 

6389 

6443 

6497 

6551 

6(504 

6058 

6712 

6760 

6820 

54 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

g 

7411 

7465 

7519 

7573 

7(52(5 

7680 

7734 

7787 

7841 

7895 

54 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

908539 

J08592 

90804(5 

908099 

908753 

908807 

908860 

908914 

908967 

54 

1 

9021 

9074 

9128 

9181 

9335 

9289 

934-' 

9390 

9449 

9503 

54 

2 

8356 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

910037 

53 

3 

9J0091 

910144 

J10197 

910251 

910304 

910358 

9101  1  1 

9104(54 

910518 

0.-.71 

53 

4 

0024 

0(578 

0731 

07H4 

0838 

0891 

0944 

0998 

1051 

1104 

53 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1177 

1530 

1584 

1037 

53 

6 

1690 

1743 

1797 

1850 

1903 

1950 

2009 

2003 

2110 

2109 

53 

7 

2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2047 

2700 

53 

8 

27531  2806 

2859 

2913 

2906 

3019 

3072 

3125 

3178 

3231 

53 

9 

32841  3337 

3390 

3443 

3496 

3549 

3(502 

3055 

3708 

3761 

53 

No.  I     0      |     1 


|     3 


5     |     6     |     7     I     8     |    0 


LOGARITHMS  OF  NUMBERS. 


193 


»|     0 


2     |     3     |     4:     |     5 


I     7     I     8     |     9 


820 

913814 

913867 

913920  913973 

914026 

914079 

914132 

914184 

914237 

914290 

1 

4343 

4396 

4449 

4502 

4555 

4(508 

4(5(50 

4713 

4766 

4819 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

3 

54(H) 

5453 

5505 

5558 

5611 

5664 

5716 

57(59 

5822 

5875 

4 

5927 

5980 

6033 

(5085 

6138 

6191 

(5243 

6296 

6349 

6401 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

7 

750G 

7558 

7611 

7663 

771(5 

7768 

7820 

7873 

7925 

7978 

8 

8030 

•  8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

918549 

1 

9001 

9653 

97061  9758 

98  10 

9862 

9914 

9967 

920019 

920071 

2 

920123 

920176 

920228  ,920280 

920332 

920384 

92043(5 

920489 

0541 

0593 

3 

OG45 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

4 

116G 

1218 

1270 

13-2^2 

1374 

1426 

1478 

1530 

1582 

1634 

5 

I<i86 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

2362 

2414 

246(5 

2516 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

8 

3-244 

3-29C. 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

42-28 

840 

924279 

924331 

924383 

924434 

924486 

924538 

924589 

944641 

924693 

924744 

1 

4796 

4848 

4899 

4951 

5003 

5051 

5106 

5157 

5209 

52(51 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5«  521 

5673 

5725 

5776 

3 

5H28 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

G342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

71(55 

7216 

72(58 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7027 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

839(5 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

850 

929419 

929470 

929521 

929572  929623 

929674 

929725 

929776 

929827 

929879 

1 

9930 

9981 

930032  930083 

930134,930185 

930236 

930287 

930338 

930389 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

16(51 

1712 

1763 

1814 

1865 

1915 

5 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

7 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

8 

3487 

3538 

3589 

3(539 

3690 

3740 

3791 

3841 

38!J2 

3943 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

860 

934498 

934549 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

2 

5507 

5558 

5608 

5(558 

5709 

5759 

5809 

5860 

5910 

59(50 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

64(53 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

5 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

70(58 

7718 

7769 

7819 

7869 

7919 

7969 

7 

8019 

8069 

8119 

81(59 

8219 

8269 

8320 

8370 

8420 

8-170 

8 

8520 

8570 

3620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

939519 

939569 

939619 

939(569 

939719 

939769 

839819 

939869 

939918 

939968 

1 

94C018 

94{MX58 

940118 

9401*58 

940218  940267 

940317 

940367 

940417 

9404CT 

2 

0516 

0566 

0610 

06(56 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

10(54 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

15  Jl 

1501 

1611 

1(5(50 

1710 

17(50 

1809 

1S59 

1909 

1958 

5 

2008 

2058 

2107 

2157 

2-207 

2250 

2306 

2355 

2405 

2455 

8 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

7 

31)011 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3993 

3643 

3(592 

3742 

379  J 

3841 

3890 

3939 

91  3989 

4038 

4088 

4137 

418(5 

4236 

4285 

4335 

4384 

4433 

No.  |     O 


|     5 


I     7     I     8     |    9     'DC 


194 


LOGARITHMS  OF  NUMBERS. 


Ha|     O 


5     I     6     I     7     I     8     I     9     iDiff. 


881  i 

944483 

944532 

944581 

944031 

944680 

944729 

944779  944828 

944877  9449271  49 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49 

2 

5409 

5518 

5567 

5616 

5605 

5715 

5764 

5813 

58(52 

5912 

49 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6356 

6305 

6354 

6403 

49 

4 

6452 

6501 

6551 

6000 

6649 

6698 

6747 

679(5 

6845 

6894 

46 

5 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

733(5 

7385 

49 

(i 

7434 

7483 

7.~>:t2 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8260 

8315 

8364 

49 

8 

8413 

8462 

8511 

8560 

8609 

8657 

8700 

8755 

8804 

8853 

49 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 
1 

949390 

9878 

949439 
9926 

949488 
9975 

949530 
950024 

949585 
950073 

949634 
950121 

949083 
950170 

949731 
950219 

949780 
9502(57 

949829 
95031(5 

49 
49 

2 

9503(55 

950414 

950402 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

'49 

s 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2103 

2211 

22(50 

48 

f, 

2308 

2356 

2405 

£453 

2502 

2550 

2599 

2047 

2696 

2744 

43 

7 

2792 

2841 

2889 

2938 

2980 

3034 

3083 

3131 

3180 

3228 

48 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3015 

36(53 

3711 

48 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

954628 

954677 

48 

1 

4725 

4773 

4821 

4869 

491S 

4966 

5014 

5002 

51  10 

5158 

48 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

4 

6168 

6216 

0205 

6313 

0301 

6409 

6457 

6505 

6553 

6601 

48 

5 

6049 

6(i97 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6 

7128 

7176 

7224 

7272 

7320 

736K 

7416 

7464 

7512 

7559 

48 

7 

7007 

7655 

7703 

7751 

77-J'J 

7847 

7894 

7942 

7990 

80:18 

48 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8408 

8516 

48 

9 

8564 

8012 

8059 

8707 

8755 

8803 

8850 

8898 

8940 

8994 

48 

9  JO 

959041 

959089 

959137 

959185 

959232 

959280 

959328 

959375 

959423 

959471 

48 

] 

9518 

9500 

9014 

9061 

9709 

9757 

9804 

9852 

9900 

9947 

48 

2 

9995 

960042 

900090 

960138 

91)0185 

960233 

9602HJ 

96032H 

960370 

900423 

48 

3 

900471 

0518 

0500 

0013 

0601 

0709 

0750 

0804 

0851 

0899 

48 

4 

0940 

OUM 

1041 

10S9 

1130 

1184 

1231 

1279 

1320 

1374 

48 

5 

1421 

1469 

1516 

1563 

1011 

1058 

1700 

1753 

1801 

1848 

47 

c 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

7 

2369 

2417 

2404 

2511 

2559 

2000 

2053 

2701 

2748 

2795 

47 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3120 

3174 

3221 

32(58 

47 

i) 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3093 

3741 

47 

920 

963788 

963835 

963882 

963929 

963977 

964024 

964071 

964118 

964165 

964212 

47 

] 

42tK) 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4037 

4084 

47 

0 

4731 

4778 

4825 

4872 

4919 

4960 

5013 

5001 

5108 

51.55 

47 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5025 

47 

4 

5072 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

604H 

6095 

47 

5 

C142 

6189 

6236 

G2S3 

632J 

6376 

6423 

6470 

6517 

(55(54 

47 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

698(5 

7033 

47 

7 

7080 

7127 

7173 

7220 

7207 

7314 

7361 

7408 

7454 

7501 

47 

8 

7548 

7595 

7042 

7688 

7735 

7782 

7829 

7875 

7922 

7909 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8430 

47 

930 
j 

968483 
8950 

968530 
8996 

968576 
9043 

968623 
9090 

968670 
9136 

968716 
9183 

968763 
9229 

968810 
9276 

968856 
9323 

968903 
9369 

47 
47 

9416 

9403 

9509 

9556 

9002 

9(549 

9Ji!)5 

9742 

9789 

9835 

47 

3 

9882 

9928 

9975 

970021 

970008 

9701  14 

970161 

J70207 

970254 

970300 

47 

4 

970347 

970393 

970440 

0480 

0533 

0579 

0020 

0072 

0719 

0765 

46 

5 

U8J2 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46 

6 

1276 

1322 

1309 

1415 

1461 

1508 

1554 

1(50] 

1647 

IfiJM 

46 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2004 

2110 

2157 

46 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

9 

2666 

2712 

2758 

2804 

2851 

2897 

29431  2989 

3035 

3082 

46 

No.)     0     |     1     i 


13|4|5|6|7|8|&|I>* 


LOGARITHMS  OF  NUMBERS. 


195 


Na|     6 


|     3     | 


|     7 


940 

973128,973174  )73220  )73206 

973313 

9733,39 

973405 

973451 

973497  973543 

46 

J 

3590 

3636 

368-2 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

4 

4972 

5018 

5064 

5110 

515C 

5202 

5248 

5294 

5340 

5386 

46 

5 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

6 

5891 

5937 

5983 

6029 

(5075 

6121 

6167 

6212 

6258 

(5304 

46 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

9 

7206 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

977769 

977815 

977861 

977906 

977952 

977998 

978043 

978089 

J78135 

46 

J 

8181 

8226 

8272 

8317 

8363 

8409 

84.34 

8500 

8546 

8591 

46 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

4 

9548 

9594 

9639 

9(>85 

9730 

9776 

9821 

9867 

9912 

9958 

46 

5 

980003 

980049 

)80094 

980140 

(80185 

980231 

980276 

980322 

980367 

980412 

45 

6 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

982316 

982362 

J82407 

982452 

982497 

982543 

982588 

982633 

982678 

45 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

358  J 

45 

3 

3026 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

5 

4387 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

G 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

.  5786 

5830 

45 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

986817 

986861 

986906 

986951 

986996 

987040 

987085 

987130 

987175 

45 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

4 

8559 

8604 

8048 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

c 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

7 

9895 

9939 

9983 

990028 

9900/2 

990117 

990161 

990206  990250 

090294 

44 

81990339 

990383 

990428 

0472 

0516 

0561 

0005 

0650 

0694 

0738 

44 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

109*3 

1137 

1182 

44 

980 

991226 

991270 

991315 

991359 

991403 

991448 

991492 

991536 

991580 

991625 

44 

.  I 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4.381 

4625 

4669 

4713 

44 

8   4757 

4801 

4H45 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

9 

5196 

5240 

50*4 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

995635 

995679 

995723 

995767 

995811 

995854 

995898 

995942 

995986 

996030 

44 

J 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6802 

6906 

44 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7255 

7299 

7343 

44 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7698 

.  7736 

7779 

44 

5 

7823 

7867 

7910 

71)54 

7998 

8041 

8085 

8129 

8172 

8216 

44 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

8 

9131 

9174 

9218 

9261 

9305 

9348 

939-2 

9435 

9479 

9522 

44 

S 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

Mu  I     O     |     1 


[     3 


5    I    6 


8    I    9    |i>ift 


TABLE   XIII. 

LOGARITHMIC    SINES,    COSINES,    TANGENTS,    AND 
COTANGENTS. 


N.  B.  — THE  minutes  in  the  left-hand  column  of  each  page, 
increasing  downwards,  belong  to  the  degrees  at  the  top  ;  and 
those  increasing  upwards,  in  the  right-hand  column,  belong  to 
the  degrees  below. 


198        (0  Degree.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Sine    |    D.       Cosine   |  D.  |   Tnng.      D.       Cotanff. 

0 

o-oooooo 

10-000000 

o-oooooo 

Infinite. 

1 

6-4637-20 

501717 

000000 

00 

6-41537-26 

501717 

1  3-536-274 

2 

7(54756 

293485 

000000 

00 

76475(5 

293483 

235-244 

3 

940H47 

2«  18231 

0(10(1(10 

00 

940847 

208231 

059153 

4 

7-065786 

101517 

00(1000 

00 

7-065786 

1(51517 

12-934214 

5 

]  6269(5 

131968 

000000 

00 

162*596 

1319(59 

837304 

6 

241877 

111575 

9-999999 

01 

241878 

111578 

75*1-2-2 

7 

308824 

96<>53 

999999 

or 

308825 

99653 

691175 

8 

366816 

85254 

999999 

01 

366817 

85-254 

6331*3 

9 

417968 

76263 

999999 

01 

417970 

76-263 

582030 

10 

463725 

68988 

999998 

01 

463727 

68988 

536-273 

11 

7-505118 

62981 

9-999998 

01 

7-505120 

6298J 

12-494*80 

12 

542906 

57936 

999997 

01 

542909 

57933 

457091 

13 

577(5(58 

53641 

999997 

01 

577672 

53(542 

4-223-28 

14 

609853 

49938 

999996 

01 

609857 

49939 

390143 

15 

639816 

46714 

999996 

01 

639820 

46715 

3(50180 

16 

667845 

43881 

999995 

01 

6(57849 

43882 

33-2151 

17 

694173 

41372 

999995 

01 

694179 

41373 

305821 

18 

718997 

39135 

999994 

0! 

719003 

39136 

2*0997 

19 

742477 

37127 

999993 

01 

742484 

37128 

257516 

20 

764754 

35315 

999993 

01 

764761 

•35136 

235-239 

21 

7-785943 

33672 

9-999992 

01 

7-785951 

33673 

12-214049 

22 

806146 

32175 

999991 

01 

806  J  55 

32176 

193845 

23 

825451 

30805 

999990 

01 

825460 

30806 

174540 

24 

843934 

29547 

999989 

02 

843944 

29549 

15(5056 

25 

8616(52 

28388 

999988 

02 

861674 

28390 

13*32(5 

2(5 

878(595 

27317 

999988 

02 

878708 

27318 

121292 

27 

895085 

2(53-23 

999987 

02 

895099 

20325 

104901 

28 

910879 

25399 

9!  19986 

02 

910894 

25401 

0*9106 

29 

926119 

24538 

999985 

02 

9S6IU4 

24540 

0738(56 

30 

940842 

23733 

999983 

02 

940858 

23735 

059142 

31 

7-955082 

22980 

9-999982 

02 

7-955100 

22981 

12-044900 

32 

9(58870 

22-273 

99998J 

02 

9(58889 

22273 

031111 

33 

982233 

21(508 

999980 

02 

982253 

21(510 

017747 

34 

995  198 

20981 

999979 

02 

995219 

'20!  1*3 

004781 

35 

8-007787 

20390 

999977 

02 

8-007*09 

20392 

11-99-2191 

36 

020021 

19831 

999976 

02 

020045 

19833 

979955 

37 

031919 

19302 

999975 

02 

03  lit  15 

19305 

9(58055 

38 

043501 

18801 

999973 

02 

043527 

1*803 

95(5473 

39 

054781 

18325 

999972 

02 

054809 

18327 

945191 

40 

065776 

17872 

999971 

02 

065806 

17874 

934194 

41 

8-07(5500 

17441 

9-999969 

02 

8-070531 

17444 

11-923469 

42 

0869(55 

17031 

999968 

02 

08(5997 

17034 

913003 

43 

097183 

1(5(539 

999986 

02 

097-217 

16642 

90-27*3 

44 

1071(57 

162(55 

9999(54 

03 

107-202 

16-2(58 

892797 

45 

1J6926 

15908 

999903 

03 

1  169(53 

15910 

8*3037 

46 

J  26471 

155(56 

999961 

03 

126510 

155(58 

873490 

47 

135*10 

15238 

999959 

0.) 

135*51 

15-241 

864149 

48 

1441153 

14924 

999958 

03 

14499(5 

149-27 

855004 

49 

153907 

14622 

99995(5 

03 

153952 

14(527 

846048 

50 

162681 

14333 

999954 

03 

162727 

J4336 

837273 

51 

8-171280  ' 

14054 

9-999952 

03 

8-171328 

14057 

11-828672 

52 

179713 

13786 

999950 

03 

1797(53 

13790 

820237 

53 

187985 

13529 

999948 

03 

18*036 

13532 

811964 

54 

196102 

13280 

999946 

03 

196156 

13284 

803844 

55 

204070 

13041 

999944 

03 

2041-2(5 

13044 

795874 

56 

211*95 

12810 

999942 

04 

'211953 

1-2*14 

7HH047 

57 

219581 

12587 

999940 

04 

219641 

12590 

7*0359 

58 

2-27134 

1-237-2 

999938 

04 

227195 

1-237(5 

772805 

59 

234557 

12164 

99993(5 

04 

234621 

1-21(18 

765379 

60 

241855 

11963 

999934 

04 

241921 

1  1967 

75*079 

|       Cosine 


Sine         | 


Cotang.      | 


)       Tang.       |  M. 


89  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (1  Degree.)        199 


M. 

Sine       D.      Cosine     D.   |   Tang.      D.      Golan*.   | 

0 

8-241855 

11963 

9-999934 

04 

8-241921 

11967 

11-758079 

1 

249033 

1  1768 

999932 

04 

249102 

11772 

750898 

2 

256094 

11580 

999!»29 

04 

256165 

11584 

743835 

3 

203042 

11398 

999927 

04 

2f)31  15 

11402 

736885 

4 

269881 

11221 

999925 

04 

269956 

11225 

730044 

5 

276614 

11050 

999922 

04 

276691 

11054 

723309 

6 

233243 

10883 

999920 

04 

283323 

10887 

716677 

289773 

10721 

999918 

04 

289856 

10726 

710144 

8 

296207 

10565 

999915 

04 

296292 

10570 

703708 

9 

302546 

10413 

999913 

04 

302634 

10418 

697366 

]<> 

308794 

10266 

999910 

04 

308884 

10270 

691116 

11 

8-314954 

10122 

9-999907 

04 

8-315046 

10126 

11-684954 

19 

321027 

9982 

999905 

04 

321122 

9987 

678878 

13 

327016 

9847 

999902 

04 

327114 

9851 

672886 

14 

332924 

9714 

999899 

05 

333025 

9719 

666975 

IS 

338753 

9586 

999897 

05 

338856 

9590 

661144 

16 

344504 

9460 

999894 

05 

344610 

9465 

655390 

17 

350181 

9338 

999891 

05 

350289 

9343 

649711 

18 

355783 

9219 

999888 

05 

355895 

9224 

644105 

19 

361315 

9103 

999885 

05 

361430 

9108 

638570 

20 

366777 

8990 

999882 

05 

366895 

8995 

633105 

21 

8-372171 

8880 

9-999879 

05 

8-372292 

8885 

11-627708 

22 

377499 

8772 

999876 

05 

377622 

8777 

622378 

23 

382762 

8667 

999873 

05 

382889 

8672 

617111 

24 

387962 

8564 

999870 

05 

388092 

8570 

611908 

SS 

393101 

8464 

999867 

05 

393234 

8470 

606766 

26 

398179 

8366 

999864 

05 

398315 

8371 

601685 

27 

403199 

8271 

999861 

05 

403338 

8276 

596662 

28 

408161 

8177 

999858 

05 

408304 

8182 

591696 

29 

413068 

8086 

999854 

05 

413213 

8091 

586787 

30 

417919 

7996 

999851 

06 

418068 

8002 

581932 

31 

8-422717 

7909 

9-999848 

06 

8-422869 

7914 

11-577131 

32 

427462 

7823 

999844 

06 

427618 

7830 

572382 

33 

432156 

7740 

999841 

06 

432315 

7745 

567685 

34 

436800 

7657 

999838 

06 

436962 

7663 

563038 

35 

441394 

7577 

999834 

06 

441560 

7583 

558440 

3f> 

445941 

7499 

999831 

06 

446110 

7505 

553890 

37 

450440 

7422 

999827 

06 

450613 

7428 

549387 

38 

454893 

7346 

999823 

06 

455070 

7352 

544930 

39 

459301 

7273 

999820 

06 

459481 

7279 

540519 

40 

463665 

7200 

999816 

06 

463849 

7206 

536151 

41 

8-467985 

7129 

9-999812 

06 

8-468172 

7135 

11-531828 

42 

472263 

7060 

999809 

06 

472454 

7066 

527546 

43 

476498 

6991 

999805 

06 

476693 

6998 

523307 

44 

480693 

6924 

999801 

06 

480892 

6931 

519108 

45 

484848 

6859 

999797 

07 

485050 

6865 

514950 

41) 

488963 

6794 

999793 

07 

489170 

6801 

510830 

47 

493040 

6731 

999790 

07 

493250 

6738 

506750 

4H 

497078 

6669 

999786 

07 

497293 

6676 

502707 

49 

501080 

6608 

999782 

07 

501298 

6615 

498702 

50 

505045 

6548 

999778 

07 

505267 

6555 

494733 

51 

8-508974 

6489 

9-999774 

07 

8-509200 

6496 

11-490800 

52 

512867 

6431 

999769 

07 

513098 

6439 

486902 

53 

516726 

6375 

999765 

07 

516961 

6382 

483039 

54 

520551 

6319 

999761 

07 

520790 

6326 

479210 

55 

524343 

6264 

999757 

07 

524586 

6272 

475414 

50 

528102 

6211 

989753 

07 

528349 

6218 

471651 

57 

531828 

6158 

999748 

07 

532080 

6165 

467920 

58 

535523 

6106 

999744 

07 

535779 

6113 

464221 

59 

539186 

6055 

999740 

07 

539447 

6062 

460553 

GO 

542819  '  6004 

999735 

07 

543084 

6012 

456916 

|      Cosine      | 


|        BUM 


Cotang.     i 


I        Tang. 


88  Degrees. 


200      (2  Degrees.)     LOGARlTmilC  SINES,  COSINES,  ETC. 


M.  |    Sine       D.   |   Cosine     D.   |   Tan?.   |   D.      Cotan?.   | 

0 

8-542819 

6004 

9-999735 

07 

8-543084 

6012 

11  45(i916 

(H) 

1 

54G422 

5955 

999731 

07 

546691 

5962 

453309 

59 

g 

549995 

5906 

999796 

07 

550268 

5914 

449732 

58 

3 

55X539 

5858 

91)9722 

08 

553817 

5866 

446183 

57 

4 

557054 

5811 

999717 

08 

557336 

5819 

442664 

56 

5 

560540 

57G5 

999713 

08 

56082S 

5773 

439172 

55 

6 

563999 

5719 

999708 

08 

564291 

5727 

435709 

54 

7 

567431 

5674 

990704 

08 

567727 

5682 

432273 

53 

8 

570836 

5630 

999699 

08 

571137 

5638 

428863 

52 

9 

574214 

5587 

999694 

08 

574520 

5595 

425480 

51 

W 

5/  <566 

5544 

999689 

08 

577877 

5552 

422123 

50 

11 

8-580892 

5OJ2 

9-999685 

08 

8-581208 

5510 

11-418792 

49 

12 

584193 

5460 

999680 

08 

584514 

5468 

.415486 

48 

13 

587469 

5419 

939075 

08 

587795 

5427 

412205 

47 

14 

590721 

5379 

999070 

08 

591051 

5387 

408949 

46 

15 

593948 

5339 

999665 

08 

594283 

5347 

4057J7 

45 

16 

597152 

5300 

999660 

08 

597492 

5308 

402508 

44 

17 

600332 

5261 

999655 

08 

600077 

5270 

399323 

43 

18 

603489 

5223 

999650 

08 

603839 

5232 

396361 

42 

19 

606623 

5186 

999645 

09 

606978 

5194 

393022 

41 

20 

609734 

5149 

999640 

09 

610094 

5158 

389906 

40 

21 

8-612823 

5112 

9-999635 

09 

8-613189 

5121 

11-386811 

3D 

°2 

615891 

5076 

999629 

09 

616262 

5085 

383738 

38 

23 

618937 

5041 

999624 

09 

619313 

5{»50 

380687 

37 

24 

621962 

5006 

999619 

09 

622343 

5015 

377(557 

36 

25 

624965 

4972 

999614 

09 

625352 

4981 

374648 

35 

26 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34 

27 

630911 

4904 

999603 

09 

631308 

4913 

368C92 

33 

28 

633854 

4871 

999597 

09 

634256 

4880 

365741 

32 

29 

(536776 

4839 

999592 

09 

6371*4 

4848 

362816 

31 

3') 

639680 

4806 

999586 

09 

640093 

4816 

359907 

30 

31 

8  642563 

4775 

9-999581 

09 

8-642982 

4784 

11-357018 

29 

32 

645428 

4743 

999575 

09 

645853 

4753 

354147 

28 

33 

648274 

4712 

999570 

09 

648704 

4722 

351296 

27 

34 

651102 

4682 

999564 

09 

651537 

4691 

348163 

26 

35 

65391  1 

4652 

999558 

10 

654:152 

4661 

345(548 

25 

36 

656702 

4622 

999553 

10 

657149 

4631 

342851 

24 

37 

659475 

4592 

999547 

10 

659928 

4602 

340072 

23 

38 

662230 

4563 

999541 

10 

662689 

4573 

33731  1 

22 

39 

664968 

4535  . 

999535 

10 

665433 

4544 

334567 

21 

40 

667689 

4506 

999529 

10 

668160 

4526 

331840 

20 

41 

8-670393 

4479 

9-999524 

10 

8-670870- 

4488 

11-329130 

19 

42 

673080 

4451 

999518 

10 

673563 

4461 

326437 

18 

43 

675751 

4424 

999512 

10 

676239 

4434 

323761 

17 

44 

678405 

4397 

999506 

10 

678900 

4417 

32  J  100 

16 

45 

681043 

4370 

999500 

10 

681544 

4380 

318456 

15 

40 

683665 

4344 

999493 

10 

684172 

4354 

315838 

14 

47 

686272 

4318 

999487 

10 

686784 

4328 

313216 

13 

48 

688863 

4292 

999481 

10 

689381 

4303 

310619 

12 

49 

691438 

4267 

999475 

10 

691963 

4277 

308037 

11 

50 

693998 

4242 

999469 

10 

694529 

4252 

3(15471 

JO 

51 

8-696543 

4217 

9-999463 

11 

8-697081 

4228 

11-302919 

9 

52 

699073 

4192 

999456 

11 

699617 

4203 

300383 

8 

53 

701589 

4168 

999450 

11 

702139 

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2978(51 

7 

54 

704090 

4144 

999443 

11 

704646 

4155 

295354 

6 

55 

706577 

4121 

999437 

11 

707140 

4132 

2928(50 

5 

56 

709049 

4097 

999431 

11 

709618 

4108 

290382 

4 

57 

711507 

4074 

999424 

11 

712083 

4085 

287917 

3 

58 

713952 

4051 

999418 

11 

714534 

4062 

285465 

2 

59 

71(5383 

4029 

9!  (9411 

11 

7l(i972 

4040 

2H3028 

] 

60  ' 

718800 

4006 

999404 

11 

719396 

4017 

280604 

0 

|       Cosine       1 


I 

87  Degrees. 


Cola.,-. 


I         Tang. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (3  Degrees.)      201 


M.    I         Sine       |         D.       |       Cosine      |     D.     |      Tang.       |        D.        |      Cotnnp.      | 


1) 

8-718800 

4006 

9-999404 

11 

8-719396 

4017 

11-280004   60 

1 

721204 

3984 

999398 

11 

721806 

3995 

'  278194   59 

o 

723.11)5 

3962 

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11 

724204 

3974 

275796 

58 

3 

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3941 

999384 

11 

720588 

3952 

273412 

57 

4 

723337 

3919 

999378 

11 

728959 

3930 

271041 

56 

5 

731)033 

3898 

999371 

11 

731317 

3909 

268083 

55 

6 

733027 

3877 

999304 

12 

733063 

3889 

2G6337 

54 

7 

735354 

3857 

999357 

12 

735996 

3868 

264004 

53 

8 

737GG7 

3836 

999350 

12 

738317 

3848 

201083 

52 

9 

73'J9:;9 

3816 

999343 

12 

740026 

3827 

259374 

51 

10 

7422-59 

3796 

999336 

12 

742922 

3807 

257078 

50 

Jl 

8-744536 

3776 

9-9993-29 

12 

8-745207 

3787 

11.254793 

49 

12 

746802 

3756 

999322 

12 

747479 

3768 

252521 

48 

13 

749055 

3737 

999315 

12 

749740 

3749 

250200 

47 

14 

751297 

3717 

999308 

12 

751989 

3729 

248011 

46 

15 

753528 

3098 

999301 

12 

754227 

3710 

245773 

45 

16 

755747 

3079 

999294 

12 

750453 

3092 

243547 

44 

17 

757955 

3001 

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12 

758668 

3073 

241332 

43 

18 

700151 

3642 

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12 

760872 

3655 

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42 

19 

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12 

763065 

3636 

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20 

764511 

3606 

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12 

765246 

3618 

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40 

21 

8-700075 

3588 

9-999257 

12 

8-71.7417 

3600 

11-232583 

39 

22 

708828 

3570 

999250 

13 

709578 

3583 

230422 

38 

23 

770970 

3553 

999242 

13 

771727 

3565 

228273 

37 

24 

773101 

3535 

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13 

773866 

3548 

226134 

36 

25 

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3518 

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13 

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3531 

224005 

35 

26 

777333 

3501 

999220 

13 

778114 

3514 

221886 

34 

27 

779434 

3484 

999212 

13 

780222 

3497 

219778 

33 

28 

781524 

3407 

999205 

13 

782320 

3480 

217680 

32 

29 

783605 

3451 

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13 

784408 

3404 

215592 

31 

30 

735G75 

3431 

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13 

786486 

3447 

213514 

30 

31 

8-787736 

3418 

9-999181 

13 

8-788554 

3431 

11-211440 

29 

32 

789787 

3402 

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13 

7900  J  3 

M14 

209387 

28 

33 

791828 

3386 

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13 

7926G2 

3399 

207338 

27 

34 

793859 

3370 

999158 

13 

794701 

3383 

205299 

26 

35 

795881 

3354 

999150 

13 

790731 

3368 

203209 

25 

3(5 

797894 

3339 

999142 

13 

798752 

3352 

201248 

24 

37 

799897 

3323 

999134 

13 

800703 

3337 

199237 

23 

38 

801892 

3308 

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13 

802765 

3322 

197235 

22 

39 

803876 

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13 

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3307 

195242 

21 

40 

805852 

3278 

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13 

800742 

3292 

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20 

41 

8-807819 

3263 

9-999102 

13 

8-808717 

3278 

11-191283 

19 

42 

809777 

3249 

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14 

810083 

3202 

189317 

18 

43 

811726 

3234 

999086 

14 

812641 

3248 

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17 

44 

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3219 

999077 

14 

814589 

3233 

185411 

16 

45 

815599 

3205 

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14 

816529 

3219 

183471 

15 

46 

817522 

3191 

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14 

818461 

3205 

1H1539 

14 

47 

819436 

3177 

999053 

14 

820384 

3191 

179010 

13 

48 

821343 

3103 

999044 

14 

822298 

3177 

177702 

12 

49 

823240 

3149 

999036 

14 

824205 

3163 

175795 

11 

50 

825130 

3135 

999027 

14 

826103 

3150 

173897 

10 

51 

8-827011 

3122 

9-999019 

14 

8-827992 

3136 

11-172008 

9 

52 

828884 

3108 

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14 

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3123 

170120 

8 

53 

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14 

831748 

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108252 

7 

54 

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3082 

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14 

833013 

3096 

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6 

55 

834450 

3009 

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14 

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3083 

164529 

5 

56 

830297 

3050 

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14 

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3070 

162(579 

4 

57 

838130 

3043 

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15 

839103 

3057 

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3 

58 

839956 

3030 

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15 

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3045 

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2 

59 

841774 

3017 

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15 

842825 

3032     157175 

1 

60 

843585 

3000 

998941 

15 

844644 

3019  1   155356 

0 

|        Cosine     J 


Tang.         |     M. 


80  Degrees. 


202      (4  Degrees.)    LOGARITHMIC  SIXES,  COSINES*  ETC. 


M. 

|    Sine 

D. 

|   Cosine 

I). 

Tan£. 

1   D. 

Cotang. 

U 

8-843585 

3005 

9-998941 

15 

8-844044 

3D  19 

11-155356 

60 

1 

845387 

2992 

998932 

15 

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3007 

153545 

59 

a 

847183 

2980 

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15 

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151740 

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3 

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15 

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2982 

149943 

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4 

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2955 

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56 

5 

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1410(58 

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9 

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51 

10 

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15 

802433 

2330 

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50 

11 

8-863014 

2873 

9-998841 

15 

8-804173 

2888 

11135827 

49 

1-2 

864738 

2801 

998832 

15 

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2877 

134094 

48 

13 

860455 

2850 

998823 

16 

837032 

2300 

1323G8 

47 

14 

8681  05 

2839 

998813 

1G 

8(39351 

2854 

130649 

46 

15 

869HG8 

2828 

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1G 

8710G4 

2343 

128936 

45 

16 

871505 

2817 

998795 

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2832 

127230 

44 

37 

873255 

2800 

9D8785 

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2821 

125531 

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18 

874938 

2795 

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42 

n 

87GG15 

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20 

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2773 

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2703 

9-998747 

10 

8-881202 

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11-118798 

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2-2 

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2752 

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10 

882809 

2708 

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38 

215 

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2742 

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10 

884530 

2758 

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37 

24 

884903 

2731 

998718 

10 

880  185 

2747 

113815 

36 

2.1 

88(3542 

2721 

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1G 

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2737 

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35 

28 

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2711 

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16 

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2727 

1  10524 

34 

27 

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2700 

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16 

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2717 

108888 

33 

28 

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16 

892742 

2707 

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32 

29 

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17 

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2697 

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31 

30 

894643 

2G70 

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17 

895984 

2087 

10401G 

30 

31 

8-896246 

2000 

9-998G49 

17 

8-897596 

2677 

11-102404 

29 

32 

897842 

2051 

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17 

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28 

33 

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2041 

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2058 

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27 

34 

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2031 

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7 

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2048 

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26 

35 

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2622 

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7 

903987 

2038 

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25 

36 

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2012 

998599 

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2029 

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24 

37 

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7 

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2020 

092853 

23 

38 

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22 

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2001 

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40 

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17 

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20 

41 

8-911949 

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9-998548 

17 

8-913401 

2583 

11-080599 

19 

42 

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998537 

17 

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2574 

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18 

43 

915022 

2547 

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17 

91(3495 

2505 

083505 

17 

44 

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16 

45 

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2529 

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18 

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15 

46 

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2520 

998495 

18 

921096 

2538 

078904 

14 

47 

921103 

2512 

998485 

18 

922019 

2530 

077381 

13 

48 

922010 

2503 

998474 

18 

924130 

2521 

0758(34 

12 

4!) 

924112 

2494 

9984G4 

18 

925649 

25  J  2 

074351 

11 

50 

925G09 

2480 

998453 

18 

92715G 

2503 

072844 

10 

51 

8-927100 

2477 

9-998442 

18 

8-928058 

2495 

11-071342 

9 

52 

928587 

24(59 

998431 

18 

930155 

248G 

069845 

8 

53 

930008 

2400 

998421 

18 

931047 

2478 

068353 

7 

54 

931544 

2452 

998410 

18 

933134 

2470 

OG68GG 

6 

5.") 

933015 

2443 

998399 

18 

934G1G 

24G1 

005384 

5 

56 

934481 

2435 

998388 

18 

930093 

2453 

003907 

4 

57 

935942 

2427 

998377 

18 

9375G5 

2445 

OG2435 

3 

58 

937398 

2419 

9983GG 

18 

939032 

2437 

000908 

2 

59 

938850 

2411 

998355 

18 

940494 

2430 

059500 

1 

GO 

94029G 

2403 

998344 

18 

941952 

2421 

058048 

0 

|      Cosine 


Sine         |  |      Coding. 

85  Degrees 


|       Tang.       |     1C, 


LOGARITHMIC  SINES,  COSINES,  ETC.    (5  Degrees.)      203 


M.     Sine       D.       Cosine   |   D.  |   Tang.      L>.     Cotanj?.   ! 

0 

8-940296 

2403 

9-998344 

19 

8-941952 

2421 

11  058048 

60 

1 

1)4)738 

2394 

9!  18333 

19 

943404 

2413 

05651  K> 

59 

2 

943174 

2387 

l)>)83-.>2 

19 

944852 

2405 

055148 

58 

3 

944006 

2379 

99831  1 

19 

946295 

2397 

053705 

57 

4 

94GOS4 

2371 

998300 

19 

947734 

2390 

052206 

56 

5 

947456 

2363 

998289 

19 

949168 

2382 

050832 

55 

6 

948874 

2355 

998277 

19 

950597 

2374 

049403 

54 

7 

950287 

2348 

998266 

19 

952021 

2366 

047!»79 

53 

8 

951696 

2340 

998255 

19 

953441 

2360 

046559 

52 

9 

953100 

2332 

998243 

19 

95485(5 

2351 

045144 

51 

10 

954499 

23-25 

998232 

19 

951)267 

2344 

043733 

53 

11 

8-9558D4 

2317 

9-998220 

19 

8-957674 

2337 

11042326 

49 

12 

957284 

2310 

998209 

19 

959075 

2329 

040925 

48 

13 

958670 

2302 

91)8197 

19 

960473 

2323 

039527 

47 

14 

960052 

2295 

9118186 

19 

961866 

2314 

038134 

46 

15 

961429 

2288 

998174 

19 

963255 

2307 

036745 

45 

16 

962801 

2280 

998163 

19 

964639 

2300 

035361 

44 

n 

964170 

2273 

998151 

19 

966019 

2293 

033981 

43 

J8 

965534 

2266 

998139 

20 

967394 

2286 

032606 

42 

19 

966893 

2259 

998128 

20 

968766 

2279 

031234 

41 

20 

968249 

2252 

998116 

20 

970133 

2271 

029867 

40 

21 

8-969600 

2244 

9-99H104 

20 

8-971496 

2265 

11028504 

39 

93 

970947 

2238 

998092 

20 

972855 

2257 

027145 

38 

23 

972289 

2231 

998080 

20 

974209 

2251 

025791 

37 

24 

973628 

2224 

998068 

20 

975560 

2244 

024440 

36 

25 

974962  . 

2217 

998056 

20 

976906 

2237 

023094 

35 

2(i 

976293 

2210 

998044 

20 

978248 

2230 

021752 

34 

27 

977619 

2203 

998032 

20 

979586 

2223 

020414 

33 

28 

978941 

2197 

998020 

20 

980921 

2217 

019079 

32 

29 

980259 

2190 

998008 

20 

982251 

2210 

017749 

31 

30 

981573 

2183 

997996 

20 

983577 

2204 

016423 

30 

31 

8-982883 

2177 

9-997984 

20 

8-984899 

2197 

11-015101 

29 

32 

984189 

2170 

997972 

20 

98(5217 

2191 

013783 

28 

33 

985491 

2163 

997959 

20 

987532 

2184 

012468 

27 

34 

986789 

2157 

997947 

20 

988842 

2178 

011158 

26 

33 

988083 

2150 

997935 

21 

990149 

2171 

009851 

25 

36 

989374 

2144 

997922 

21 

991451 

2165 

008549 

24 

37 

990660 

2138 

997910 

21 

992750 

2158 

007250 

23 

38 

991943 

2131 

997897 

21 

994045 

2152 

005955 

22 

39 

993222 

2125 

997885 

21 

995337 

2146 

004063 

21 

40 

994497 

2119 

997872 

21 

996624 

2140 

003376 

20 

41 

8-995768 

2112 

9-997860 

21 

8-997908 

2134 

11-002092 

19 

42 

997036 

2106 

997847 

21 

9991  SB 

2127 

000812 

]8 

43 

998299 

2100 

997835 

21 

9-000465 

2121 

10-999535 

n 

44 

999560 

2094 

997822 

21 

001738 

2115 

998262 

16 

45 

9-000816 

2087 

997809 

21 

003007 

2109 

996993 

15 

46 

002069 

2082 

997797 

21 

004272 

2103 

995728 

14 

47 

003318 

2076 

997784 

21 

005534 

2097 

994466 

13 

48 

004563 

2070 

997771 

21 

006792 

2091 

993208 

12 

49 

0058(15 

2064 

997758 

21 

008047 

2085 

991953 

11 

50 

007041 

2058 

997745 

21 

009298 

2080 

990702 

10 

51 

J-008278 

2052 

9-997732 

21 

9-010546 

2074 

10-989454 

9 

52 

009510 

2046 

997719 

21 

011790 

2068 

988210 

8 

53 

1  010737 

2040 

997706 

21  ' 

013031 

2062 

986969 

7 

54 

01  1962 

2034 

997693 

22 

014268 

2056 

985732 

6 

55 

013182 

2029 

997680 

22 

015502 

2051 

984498 

5 

5C 

014400 

2023 

997667 

22 

016732 

2045 

983268 

4 

57 

015613 

2017 

997654 

22 

017959 

2040 

982041 

3 

58 

016824 

2012 

997641 

22 

019183 

2033 

980817 

2 

59  j  018031 

2006 

997628 

22 

020403 

2028 

979597 

1 

60  '  019235 

2000 

997614 

22 

021620 

2023 

978380 

0 

|   Cosmo   |        |    Sine    |       |   Cotang.   ;           Tang.     M. 

84  Degrees. 

204      (6  Degrees.)    LOGARITHMIC  SINES,  COSINES,  ETC. 


M.    |       Sine 


D.     |       Tang.        |        D. 


Cotang. 


0 

1 

9-019235 
020435 

2000 
1995 

9-997(514 
997(501 

22 
22 

9-021620 

022834 

2023 
2017 

10-978380 
9771(56 

2 

021032 

1989 

997588 

22 

024044 

2011 

975956 

3 

022825 

1984 

997574 

22 

025251 

2006 

974749 

4 

024016 

1978 

9975(51 

22 

020455 

2000 

973545 

5 

025203 

1973 

997547 

22 

027055 

1995 

972345 

6 

026386 

1967 

997534 

23 

028852 

1990 

971148 

7 

027567 

1962 

997520 

23 

030046 

1985 

9(59954 

8 

028744 

1957 

997507 

23 

031237 

1979 

908703 

9 

029918 

1951 

997493 

23 

032425 

1974 

967575 

10 

031089 

1947 

997480 

23 

033609 

1969 

906391 

11 

9-032257 

1941 

9-097466 

23 

9-034791 

1964 

10-965209 

12 

033421 

1936 

997452 

23 

035909 

19,58 

9(54031 

13 

034  582 

1930 

997439 

23 

037144 

1953 

962856 

14 

035741 

1925 

997425 

23 

038316 

1948 

9(51(584 

15 

036896 

1920 

997411 

23 

039485 

1943 

960515 

16 

038048 

11115 

997397 

23 

040(551 

1938 

959349 

17 

039197 

1910 

997383 

23 

041813 

1933 

958187 

18 

0411342 

1905 

997309 

23 

042973 

1928 

957027 

19 

041485 

1899 

997355 

23 

044130 

1923 

955870 

20 

042(525 

1894 

997341 

23 

045284 

1918 

954716 

21 

9  043762 

1889 

9-997327 

24 

9-046434 

1913 

10-9535(16 

22 

044895 

1884 

997313 

24 

047582 

1908 

952418 

23 

04(5026 

1879 

997299 

24 

048727 

1903 

951273 

24 

047154 

1875 

997285 

24 

049869 

1898 

950131 

25 

048279 

1870 

997271 

24 

051008 

1893 

948992 

26 

049400 

1865 

997257 

24 

052144 

1889 

947856 

27 

050519 

1860 

997242 

24 

053277 

1884 

940723 

28 

051635 

1855 

997228 

24 

054407 

1879 

945593 

29 

052749 

1850 

997214 

24 

055535 

1874 

944465 

30 

053859 

1845 

997199 

24 

056659 

1870 

943341 

31 

054966 

1841 

9-997185 

24 

9-057781 

1865 

10-942219 

32 

05(5071 

1836 

997170 

24 

058900 

1869 

941100 

33 

057172 

1831 

997156 

24 

00001(5 

1855 

939984 

34 

058271 

1827 

997141 

24 

061130 

1851 

938870 

35 

059367 

1822 

997127 

24 

002240 

1846 

937700 

36 

0604(50 

1817 

997112 

24 

0(53348 

1842 

93(5652 

37 

061551 

1813 

997098 

24 

004453 

1837 

935547 

38 

0(52(539 

1808 

997083 

25 

065556 

1833 

934444 

39 

0(53724 

1804 

997068 

25 

06(5655 

1828 

833345 

40 

0(54806 

1799 

997053 

25 

067752 

1824 

932248 

41 

9-065885 

1794 

9997039 

25 

9-068846 

1819 

10-931154 

42 

060902 

1790 

997024 

25 

009938 

1815 

9300(52 

43 

068036 

1786 

997009 

25 

07)027 

1810 

928973 

44 

069107 

1781 

996994 

25 

072113 

1806 

927887 

45 

070176 

1777 

996979 

25 

073197 

1802 

926803 

46 

071242 

1772 

996964 

25 

074278 

1797 

925722 

47 

072306 

1768 

99(5949 

25 

075356 

1793 

924644 

48 

073306 

1763 

99(5934 

25 

070432 

1789 

9235(58 

49 

074424 

1759 

99(5919 

25 

077.505 

1784 

922495 

50 

075480 

1755 

996904 

25 

078576 

1780 

921424 

51 

9-070533 

1750 

9-996889 

25 

9-079044 

1776 

10-920356 

52 

077583 

1746 

99(5874 

25 

080710 

1772 

919290 

53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

54 

079676 

1738 

996843 

25 

082833 

763 

917167 

55 

080719 

1733 

990828 

25 

083891 

759 

916109 

56 

081759 

1729 

996812 

26 

084947 

755 

915053 

57 

082797 

1725 

990797 

26 

080000 

751 

914000 

58 

083832 

1721 

996782 

26 

087050 

747 

912950 

59 

0848(54 

1717 

99(5766 

26 

088098 

743 

911902 

60 

085894 

1713 

996751 

26 

089144 

1738 

910856 

Cosine       I 


|      Cotang.     | 


I      Tang.         (  M. 


&  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (7  Degrees.)      205 


M.     Sine       D.      Cosine   |   D.   |   Tan?. 

D.   |   Coianff.   \ 

IP 

9-085894 

J713 

9-996751 

26 

9-089144 

1738 

10-910856 

60 

1 

086922 

1709 

996735 

26 

090187 

1734 

909813 

59 

2 

087947 

1704 

996720 

2(5 

091228 

1730 

908772 

58 

3 

088970 

1700 

99(5704 

26 

0922(56 

1727 

907734 

57 

4 

089990 

1696 

996(588 

2(5 

093302 

1722 

906698 

56 

5 

0910(18 

1692 

996673 

26 

094336 

1719 

9056(54 

55 

6 

092024 

1(588 

99(5657 

26 

0953H7 

1715 

904633 

54 

7 

093037 

1684 

996641 

26 

09(5395 

17JI 

903605 

53 

8 

094047 

1680 

996(525 

06 

097422 

1707 

902578 

52 

9 

095050 

HJ76 

996(510 

26 

098446 

1703 

901554 

51 

JO 

09G062 

1673 

996594 

26 

099468 

1699 

900532 

50 

11 

9-097065 

1668 

9-996578 

27 

9-100487 

1695 

10-899513 

49 

12 

098006 

1685 

996562 

27 

101504 

1691 

898496 

48 

13 

099065 

11561 

99(55-1(5 

27 

102519 

1887 

897481 

47 

14 

100062 

1657 

99(5530 

27 

103532 

1684 

89(54(58 

46 

15 

10)056 

1633 

996514 

27 

104542 

1(580 

895458 

45 

16 

1020-48 

1649 

99(5498 

27 

105550 

1676 

894450 

44 

17 

103037 

1645 

996482 

27 

106556 

1672 

893444 

43 

18 

104023 

1641 

996-1(55 

27 

107559 

1(569 

89244  1 

42 

19 

105010 

1638 

996449 

27 

1085(50 

1665 

891440 

41 

20 

105992 

1634 

996433 

27 

109559 

1661 

890441 

40 

21 

9-106973 

1630 

9-996417 

27 

9-110556 

1658 

10-889444 

39 

22 

107951 

1627 

996400 

27 

111551 

1654 

888449 

38 

23 

108927 

1623 

99(5384 

27 

1  12543 

1650 

887457 

37 

24 

109901 

1619 

996368 

27 

113533 

16-1(5 

886467 

36 

25 

1J0873 

1616 

996351 

27 

114521 

1643 

885479 

35 

26 

111842 

1612 

99(5335 

27 

115507 

1639 

884493 

34 

27 

112809 

1608 

99(5318 

27 

116491 

1636 

883509 

33 

28 

113774 

1605 

996302 

28 

117472 

1632 

882528 

32 

29 

114737 

1601 

99(5285 

28 

118452 

1629 

881548 

31 

30 

115698 

1597 

996269 

28 

119429 

1625 

880571 

30 

31 

9-116656 

1594 

9-996252 

28 

9-120404 

1622 

10-879596 

29 

32 

117613 

1590 

996235 

28 

121377 

1618 

878623 

28 

33 

11  8567 

1587 

996219 

28 

122348 

1615 

877652 

27 

34 

119519 

15B3 

996202 

28 

123317 

1611 

876683 

26 

35 

120469 

1580 

996185 

28 

124284 

1607 

875716 

25 

36 

121417 

1576 

996168 

28 

125249 

1604 

874751 

24 

37 

122362 

1573 

996151 

28 

126211 

1601 

873789 

23 

38 

123306 

1569 

996134 

28 

127172 

1597 

872828 

22 

39 

124248 

1566 

996117 

28 

128130 

1594 

871870 

21 

40 

125187 

1562 

996100 

28 

129087 

1591 

870913 

20 

41 

9-126125 

1559 

9-996083 

29 

9-130041 

1587 

10-869959 

19 

42 

127060 

1556 

996066 

29 

130994 

1584 

8(59006 

18 

43 

127993 

1552 

99(5049 

29 

131944 

1581 

868056 

17 

44 

128925 

1549 

996032 

29 

132893 

1577 

867107 

16 

45 

129854 

1545 

996015 

29 

133839 

1574 

866161 

15 

46 

130781 

1542 

995998 

29 

134784 

1571 

865216 

14 

47 

131706 

1539 

995980 

29 

135726 

1567 

864274 

13 

48 

132630 

1535 

995963 

29 

136667 

1564 

8(53333 

12 

49 

133551 

1532 

995946 

29 

137605 

1561 

8(52395 

11 

50 

134470 

1529 

995928 

29 

138542 

1558 

861458 

10 

51 

9-135387 

1525 

9-995911 

29 

9-139476 

1555 

10-860524  . 

9 

52 

1315303 

1522 

995894 

29 

140409 

1551 

859591 

8 

53 

137216 

1510 

995876 

29 

141340 

1548 

858660 

7 

54 

138128 

1516 

995859 

29 

142269 

1545 

857731 

8 

55 

139037 

1512 

995841 

29 

143196 

1542 

856804 

5 

56 

139944 

1509 

995823 

29 

144121 

1539 

855879 

4 

57 

140850 

1506 

995806 

29 

145(144 

1535 

85495(5 

3 

58 

141754 

1503 

995788 

29 

145966 

1532 

854034 

2 

59 

142655 

1500 

995771 

29 

146885 

1529 

853115 

60  1  143555 

1496 

995753 

29 

147803 

1526 

852197  1  0 

I        Sine 


Cotang-.      | 


I         Tang.       |  M. 


82  Degrees. 


206      (8  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.   |       Sme         j 


Cosine         |      D.      | 


|         D.        | 


0 

9-143555 

1496 

9-995753 

30 

9-147803 

1526 

10.852197 

60 

1 

144453 

1493 

91)5735 

30 

148718 

1523 

851  282 

59 

2 

145349 

1490 

9957.17 

30 

149(532 

1520 

850368 

58 

3 

146243 

1487 

995699 

30 

150544 

1517 

849456 

57 

4 

147136 

1484 

995(581 

30 

151454 

1514 

848546 

56 

5 

148026 

1481 

9956(54 

30 

1523(53 

1511 

847637 

55 

6 

148915 

1478 

995646 

30 

153269 

1508 

846731 

54 

7 

141)802 

1475 

995628 

30 

154174 

1505 

845826 

53 

8 

150(586 

1472 

995(510 

30 

155077 

1502 

844923 

52 

D 

151569 

1469 

995591 

30 

155978 

1499 

844022 

51 

10 

1  52451 

1466 

995573 

30 

156877 

1496 

843123 

50 

11 

9-153330 

1463 

9-995555 

30 

9-157775 

1493 

10-842225 

49 

12 

154208 

1460 

995537 

30 

158671 

1490 

841329 

48 

13 

155083 

1457 

995519 

30 

159565 

1487 

840435 

47 

14 

155957 

1!54 

995501 

31 

160457 

1484 

839543 

46 

15 

156830 

1451 

99548-2 

31 

161347 

1481 

838653 

45 

16 

157700 

1448 

995464 

31 

162236 

1479 

837764 

44 

17 

158569 

1445 

99544G 

31 

1(53123 

1476 

836877 

43 

18 

159435 

1442 

995427 

31 

164008 

1473 

835992 

42 

19 

160301 

1439 

995409 

31 

164892 

1470 

835108 

41 

20 

161164 

1436 

995390 

31 

165774 

1467 

834226 

40 

21 

9-162025 

1433 

9-995372 

31 

9'  166654 

1464 

10-833346 

39 

22 

162885 

1430 

995353 

31 

167532 

1461 

832468 

38 

23 

163743 

1427 

995334 

31 

168409 

1458 

831591 

37 

24 

164600 

1424 

995316 

31 

169284 

1455 

830716 

36 

25 

165454 

1422 

995297 

31 

170157 

1453 

829843 

35 

26 

J  (56307 

1419 

995278 

31 

171029 

1450 

828971 

34 

27 

167159 

1416 

995260 

31 

171899 

1447 

828101 

33 

28 

168008 

1413 

995241 

32 

172767 

1444 

827233 

32 

2!) 

16f<856 

1410 

995222 

32 

173634 

1442 

826366 

31 

30 

169702 

1407 

995203 

32 

174499 

1439 

825501 

30 

31 

9-170547 

1405 

9-995184 

32 

9-1753(52 

1436 

10-824638 

29 

32 

71389 

1402 

995165 

32 

17G234 

1433 

823776 

28 

33 

72230 

1399 

995146 

32 

177084 

1431 

822916 

27 

34 

73070 

1396 

995127 

32 

177942 

1428 

822058 

26 

35 

73908 

1394 

995108 

32 

178799 

1425 

821201 

25 

36 

74744 

1391 

995089 

32 

179655 

1423 

820345 

24 

37 

75578 

1388 

995070 

32 

180508 

1420 

819492 

23 

38 

764  1  1 

1386 

995051 

32 

181360 

1417 

818640 

22 

39 

77242 

1383 

995032 

32 

182211 

1415 

817789 

21 

40 

178072 

1380 

995013 

32 

183059 

1412 

81(5941 

20 

41 

9-178900 

1377 

9-994993 

32 

9-183907 

1409 

10-816093 

19 

42 

179726 

1374 

994974 

32 

184752 

1407 

815248 

18 

43 

180551 

1372 

994955 

32 

185597 

1404 

814403 

17 

44 

181374 

1369 

991935 

32 

18(5439 

1402 

8135(51 

16 

45 

182196 

1366 

9J4916 

33 

187280 

1399 

812720 

15 

46 

183016 

1364 

99189(3 

33 

188120 

139(5 

811880 

14 

47 

183834 

1361 

994877 

33 

1889.58 

1393 

811042 

13 

48 

184651 

1359 

994857 

33 

189794 

1391 

81(1206 

12 

49 

185466 

135(5 

994838 

33 

190629 

1389 

809371 

11 

50 

186280 

1353 

994818 

33 

191462 

1386 

808538 

10 

51 

9-187092 

1351 

9-994798 

33 

9-  192294 

1384 

10-807706 

9 

52 

187903 

1348 

994779 

33 

193124 

1381 

80687(5 

8 

53 

188712 

1346 

994759 

33 

193953 

1379 

80(5047 

7 

54 

189519 

1343 

9111739 

33 

194780 

1376 

805220 

6 

55 

190325 

1341 

99171  9 

33 

195606 

1374 

804394 

5 

56 

191130 

1338 

994700 

33 

196430 

1371 

803570 

4 

57 

191933 

1336 

994(580 

33 

197253 

1369 

802747 

3 

58 

192734 

1333 

994660 

33 

198074 

13(5(5 

801926 

2 

59 

193534 

1330 

994640 

33 

198894 

1364 

801106 

1 

60 

194332 

1328 

994(520 

33 

199713 

1361  ' 

800287 

0 

Cosme 


Sine          1  |      Cotang. 

81  Decrees. 


|        Tang. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (9  Degrees.)      207 


}    Sine    |    D.   |   Cosine     D.     Tan?.   |    D.     Cotansr. 

|  9-194332 

1328 

9-994(520 

33  [9-199713 

1361 

10-800287 

60 

195129 

1326 

994000 

33 

200529 

1359 

799471 

59 

195925 

1323 

!!!)4580 

33 

201345 

1356 

798055 

58 

196719 

1321 

9945(50 

34 

202159 

J354 

797841 

57 

197511 

1318 

994540 

34 

202971 

1352 

797029 

56 

198302 

1316 

994519 

34 

203782 

1349 

790218 

55 

199091 

1313 

994499 

34 

204592 

1347 

795408 

54 

199879 

1311 

994479 

34 

205400 

1345 

794(500 

53 

200(500 

1308 

994459 

34 

20(5207 

1342 

793793 

52 

201451 

1300 

994438 

34 

207013 

1340 

792987 

51 

202234 

1304 

994418 

34 

207817 

1338 

792183 

50 

0-203017 

1301 

9-994397 

34 

9-208619 

1335 

10-791:581 

49 

203797 

129!) 

994377 

34 

209420 

1333 

790580 

48 

204577 

1290 

994357 

34 

210220 

1331 

789780 

47 

205354 

1294 

994330 

34 

211018 

1328 

788982 

46 

200  I'M 

1292 

994316 

34 

211815 

1326 

788185 

45 

206906 

1289 

994295 

34 

212011 

1324 

787389 

44 

207(579 

1287 

994274 

35 

213405 

1321- 

780595 

43 

208452 

1285 

994254 

35 

214198 

1319 

785802 

42 

209222 

1282 

994233 

35 

214989 

1317 

785011 

41 

209992 

1280 

•994212 

35 

215780 

1315 

784220 

40 

9-210700 

1278 

9-994191 

35 

9-21(5508 

1312 

10-783432 

39 

211526 

1275 

994171 

35 

217350 

1310 

782044 

38 

212291 

1273 

994150 

35 

218142 

1308 

781858 

37 

213055 

1271 

994129 

35 

218926 

1305 

781074 

36 

213818 

1208 

994108  . 

35 

219710 

1303 

780290 

35 

214579 

1266 

994087 

35 

220492 

1301 

779508 

34 

215338 

1264 

994006 

35 

221272 

1299 

778728 

33 

216097 

1201 

994)45 

35 

222052 

1297 

777948 

32 

21(5854 

1259 

994024 

35 

222830 

1294 

777170 

31' 

217609 

1257 

994003 

35 

223606 

1292 

770394 

30 

9-218363 

1255 

9-993981 

35 

9-224382 

1290 

10-775018 

29 

219116 

1253 

993900 

35 

225150 

1288 

774844 

28 

219868 

1250 

993939 

35 

225929 

1286 

774071 

27 

220018 

1248 

993918 

35 

226700 

1284 

773300 

26 

221367 

1240 

993890 

30 

227471 

1281 

772529 

25 

222115 

1244 

993875 

30 

228239 

1279 

771701 

24 

222861 

1242 

993854 

30 

229007 

1277 

770993 

23 

223006 

1239 

993832 

36 

229773 

1275 

770227 

22 

224349 

1237 

993811 

36 

230539 

1273 

7(59401 

21 

225092 

1235 

993789 

36 

231302 

1271 

768(598 

20 

9-225833 

1233 

9-993708 

36 

9-232065 

1269 

10-707935 

19 

220573 

1231 

993740 

36 

232820 

1207 

707174 

18 

227311 

1228 

993725 

I»6 

233586 

1265 

760414 

17 

228048 

1226 

993703 

36 

234345 

1262 

7(55055 

16 

228784 

1224 

993081 

36 

235103 

1200 

704897 

15 

229518 

1222 

993000 

3(5 

235859 

1258 

704141 

11 

230252 

1220 

993038 

36 

230(514 

1250 

763386 

13 

230984 

1218 

993010 

36 

2373(58 

1254 

762032 

12 

231714 

1210 

993594 

37 

238120 

1252 

701880 

11 

232444 

1214 

993572 

37 

238872 

1250 

701128 

10 

9-233172 

1212 

9-993550 

37 

9-239022 

1248 

10-700378 

9 

233899 

1209 

993528 

37 

240371 

1240 

759029 

3 

234025 

1207 

993500 

37 

241118 

1244 

758882 

7 

235349 

1205 

993484 

37 

241805 

1242 

758135 

6 

236073 

1203 

993402 

37 

242010 

1240 

75739(1 

5 

23(5795 

1201 

993440 

37 

243354 

1238 

750046 

4 

237515 

1199 

993418 

37 

244097 

1230 

755903 

3 

238235 

1197 

993396 

37 

244839 

1234 

755101 

1 

238953 

1195 

993374 

37 

245579 

12"° 

754421 

1 

239670 

1193 

993351 

37 

246319 

1230 

753681 

0 

|      Coswe 


|  |      Coiang. 

80  Degrees. 


|        Tang.      I  M. 


208      (10  Degrees.)     LOGARITHMIC  SINES,    COSINES,  ETC. 


M. 

|   Sine 

J   D- 

Cosine 

1  D. 

Tan*. 

D. 

1   Cotang, 

1 

0 

9--239670 

1193 

9-993351 

37 

9246319 

1230 

10-7;j%81 

60 

1 

'24U:J86 

1191 

9933-29 

37 

247057 

1228 

752943 

59 

2 

241101 

1189 

993307 

37 

247794 

2-26 

75-221  Hi 

58 

3 

241814 

1187 

993285 

37 

248530 

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751470 

57 

4 

242526 

1185 

99321  a 

37 

249204 

•2-2-2 

750736 

56 

5 

243237 

J  l;<\ 

993-240 

37 

249998 

•2-20 

75001)2 

55 

6 

£431)47 

1181 

993-217 

38 

250730 

-218 

74SW70 

54 

7 

244656 

1179 

993195 

38 

251401 

-217 

748539 

53 

8 

245363 

1177 

993172 

38 

'252191 

-215 

747809 

52 

9 

240069 

1175 

993149 

38 

2529-20 

213 

7470HO 

51 

10 

246775 

1173 

993127 

38 

253648 

'211 

746352 

50 

11 

9-247478 

1171 

9-993104 

38 

9-254374 

1209 

10-745'.i26 

49 

12 

248181 

1169 

993081 

38 

255100 

1-207 

744900 

48 

13 

248883 

1167 

993059 

38 

255824 

1205 

744  176 

47 

14 

249583 

1165 

993036 

38 

256547 

1203 

743453 

46 

15 

250282 

1163 

993013 

38 

257-269 

1-201 

74-2731 

45 

16 

250980 

1161 

99-2990 

38 

257990 

1-200 

742010 

44 

17 

251677 

1159 

99-2967 

38 

258710 

JJ98 

741290 

43 

18 

252373 

1158 

99-2J44 

38 

259429 

1196 

740571 

42 

19 

253067 

1156 

992921 

38 

260146 

1194 

739854 

41 

20 

253761 

1154 

99-2898 

38 

260863 

119-2 

739137 

40 

21 

9-254453 

1152 

9-992875 

38 

9-261578 

1190 

10-738422 

39 

22 

255144 

1J50 

992852 

38 

262-292 

1189 

737708 

38 

23 

255834 

1148 

99-2829 

39 

263005 

na7 

730995 

37 

24 

256523 

1146 

992806 

39 

263717 

1185 

730283 

36 

25 

257211 

1144 

992783 

39 

204428 

1183 

735572 

35 

26 

257898 

1142 

992759 

39 

205138 

1181 

734802 

34 

27 

258583 

1141 

99-2736 

39 

205847 

1179 

734153 

33 

28 

259208 

1139 

992713 

39 

'26u555 

1178 

733445 

32 

29 

259951 

1137 

992690 

39 

207201 

1176 

73-2739 

31 

30 

260633 

1135 

992o66 

39 

267907 

1174 

732033 

30 

31 

9-261314 

1133 

9-992643 

39 

9-268671 

117-2 

10-731329 

29 

32 

261994 

1131 

992619 

39 

209375 

1170 

730625 

28 

33 

26-2673 

1130 

99-2596 

39 

270077 

1169 

7299'23 

27 

34 

263351 

1128 

992572 

39 

270779 

1167 

729221 

26 

35 

264027 

1126 

992549 

39 

271479 

1165 

7285-21 

25 

36 

264703 

1124 

992525 

39 

27-2178 

1164 

727822 

24 

37 

265377 

1122 

992501 

39 

272876 

1162 

7271-24 

23 

38 

266051 

1130 

992478 

40 

'273573 

1160 

726427 

22 

3!) 

266723 

1119 

992454 

40 

2M209 

1158 

725731 

21 

40 

267395 

1117 

992430 

40 

274964 

1157 

7'25036 

20 

41 

9-268065 

1J15 

9-992406 

40 

9-275658 

1155 

10-724342 

19 

42 

268734 

1113 

992382 

40 

276351 

1153 

72;>049 

18 

43 

269402 

1111 

992359 

40 

277043 

1151 

722957 

17 

44 

270069 

1110 

992335 

40 

277734 

1150 

72226U 

16 

45 

270735 

1108 

99-2311 

40 

278424 

1148 

721576 

15 

46 

271400 

1106 

992287 

40 

279113 

1147 

720887 

14 

47 

272064 

1105 

99:2263 

40 

'279801 

1115 

720199 

13 

48 

272720 

1103 

992239 

40 

2c04d8 

1143 

719512 

12 

49 

273388 

1J01 

99-2214 

40 

2rilJ74 

1141 

7188'26 

11 

50 

274049 

1099 

992190 

40 

281858 

1140 

718142 

10 

51 

J-274708 

1098 

9-99211)6 

40 

9-282542 

1138 

10-717458 

9 

52 

275367 

1096 

•J92142 

40 

283!2'25 

1136 

710775 

8 

53 

270024 

1094 

992117 

41 

'283907 

1135 

716093 

7 

54 

270681 

1092 

992093 

41 

284588 

1133 

715412 

6 

55 

277337 

1091 

99-2009 

41 

28.V208 

1131 

714732 

5 

56 

277991 

1089 

992044 

41 

285947 

1130 

714053 

4 

57 

278644 

1087 

992020 

41 

iMS024 

ll;J8 

713376 

3 

58 

279297 

108(5 

991996 

41 

'287301 

1121) 

712699 

2 

59 

279948 

1084 

991971 

41 

2871*77 

1125 

712023 

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280599 

1082 

991947 

41 

288652  . 

1123 

711348 

0 

|      Cusine 


|         8,ne        | 


|      Cotang. 


I         Tang. 


79  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (11  Degrees.)      209 


M.  |        Sine         |        D.       |      Cosine 


Tang. 


Cotang.       I 


0 

9  -28059!  » 

1082 

9-991947 

41 

9-28S652 

1123 

10-711348 

60 

1 

281248 

1081 

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41 

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1122 

710674 

59 

2 

281897 

1079 

991897 

41 

289999 

1120 

710001 

58 

3 

282,544 

1077 

991873 

41 

290671 

1118 

709329 

57 

4 

283190 

1076 

991848 

41 

291342 

1117 

708658 

56 

5 

283836 

1074 

991823 

41 

292013 

1115 

707987 

55 

6 

284480 

1072 

991799 

41 

292682 

1114 

707318 

54 

7 

2S5124 

1071 

991774 

42 

293350 

1112 

706650 

53 

8 

285766 

1069 

991749 

42 

294017 

1111 

705983 

52 

9 

28(i408 

1067 

991724 

42 

294684 

1109 

705316 

51 

10 

287048 

1066 

991699 

42 

295349 

1107 

704651 

50 

11 

9-237687 

1064 

9-991674 

42 

9-296013 

1106 

10-703987 

49 

IS 

288326 

1063 

991649 

42 

296677 

1104 

703323 

48 

13 

288964 

1061 

991624 

42 

297339 

1103 

702661 

47 

14 

289600 

1059 

991599 

42 

298001 

1101 

701999 

46 

15 

290236 

1058 

991574 

42 

298662 

1100 

701338 

45 

JO 

290870 

1056 

991549 

42 

299322 

1098 

700678 

44 

17 

291504 

1054 

991524 

42 

299980 

1096 

700020 

43 

If 

292137 

1053 

991498 

42 

300(538 

1095 

699362 

42 

19 

2927(58 

1051 

991473 

42 

301295 

1093 

698705 

41 

20 

293399 

1050 

991448 

42 

301951 

1092 

698049 

40 

21 

9-294029 

1048 

9-991422 

42 

9-302607 

1090 

10-697393 

39 

82 

2946.58 

1046 

991397 

42 

303261 

1089 

696739 

38 

2:5 

295286 

1045 

991372 

43 

303914 

1087 

696086 

37 

24 

295913 

1043 

991346 

43 

304567 

1086 

695433 

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2r, 

29(5539 

1042 

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305218 

1084 

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35 

26 

297164 

1040 

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43 

305869 

1083 

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34 

27 

297788 

1039 

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306519 

1081 

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33 

28 

298412 

1037 

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307168 

1080 

692832 

32 

89 

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1036 

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307815 

1078 

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31 

30 

299055 

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43 

308463 

1077 

691537 

30 

31 

9-300276 

1032 

9-991167 

43 

9-309109 

1075 

10-690891 

29 

32 

300895 

1031 

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43 

309754 

1074 

690246 

28 

33 

301514 

1029 

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43 

310398 

1073 

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27 

34- 

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1028 

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43 

311042 

1071 

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311685 

1070 

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25 

36 

303364 

1025 

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312327 

1068 

687673 

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303979 

1023 

991012 

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312967 

1067 

687033 

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38 

304593 

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313608 

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68(5392 

22 

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314247 

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9-306430 

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1053 

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1000 

990591 

44 

323106 

1044 

676894 

7 

54 

314297 

998 

090565 

44 

323733 

1043 

676267 

6 

5.1 

314897 

997 

990538 

44 

324358 

1041 

675642 

5 

56 

315495 

996 

990511 

45 

334983 

1040 

675017 

4 

57 

316092 

994 

990485 

45 

325(507 

1039 

674393 

3 

58 

316689 

993 

990458 

45 

326231 

1037 

673769 

o 

59 

317284 

991 

990431 

45 

326853 

1036 

673147 

1 

60  '  317879 

990 

990404 

45  ' 

327475 

1035 

672525 

0 

Couite      | 


Sine         |  I      Cotang.      | 

78  Degrees. 


I       Tang.       |    M. 


210      (12  Degrees.)    LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Bine    |   D.      Co«ne    |   D.   |    Tang.   |    D.       Cotang. 

0 

9-317879 

990 

9-990404 

45 

9-327474 

1035 

10-672526 

(50 

1 

318473 

988 

990378 

45 

328095 

1033 

671905 

59 

2 

319066 

987 

990351 

45 

328715 

1032 

G71285 

SB 

3 

319(558 

986 

990324 

45 

329334 

1030 

670(JG6 

57 

4 

320249 

984 

990297 

45 

321)953 

1029 

670047 

5(5 

5 

320840 

983 

990270 

45 

330570 

1028 

661)430   55 

6 

321430 

982 

990243 

45 

331187 

1026 

6C8813   54 

7 

322019 

980 

991,215 

45 

331803 

1025 

668197 

53 

8 

322607 

979 

990188 

45 

332418 

1024 

667582 

52 

y 

323194 

977 

91)0101 

45 

333033 

1023 

666967 

51 

10 

323780 

976 

990134 

45 

333046 

1021 

660354 

50 

n 

9-324366 

975 

9-990107 

-  46 

9-334259 

1020 

10-665741 

49 

12 

324!,'50 

973 

990075) 

46 

33487] 

1019 

665129 

48 

13 

3-2.3534 

972 

99U052 

46 

335482 

1017 

664518 

47 

14 

3261  17 

970 

990025 

46 

336093 

1016 

6631)0*7 

46 

15 

320700 

969 

981)997 

46 

336702 

1015  - 

663298 

45 

16 

327281 

968 

989970 

46 

337311 

1013 

662689 

44 

17 

327862 

966 

989942 

46 

337919 

1012 

662081 

43 

18 

328442 

965 

989915 

46 

338527 

1011 

661473 

4-> 

19 

329021 

964 

989887 

46" 

339133 

1010 

660867 

41 

20 

329599 

962 

989860 

46 

239739 

1008 

660361 

40 

21 

9-330176 

961 

9-989832 

46 

9-340344 

1007 

10-65!:  056 

39 

22 

330753 

960 

989804 

46 

340948 

1006 

651)1152 

38 

23 

331329 

958 

989777 

46 

341552 

1004 

058448 

37 

24 

331903 

957 

989749 

47 

342155 

1003 

657845 

3(5 

25 

332478 

956 

989721 

47 

342757 

1002 

657243 

35 

26 

333051 

954 

989693 

47 

343358 

1000 

656642 

34 

27 

333G24 

953 

989065 

47 

343958 

999 

656042 

33 

28 

334195 

952 

989637 

47 

344558 

998 

655442 

32 

29 

334766 

950 

989609 

47 

345157 

997 

654843 

31 

30 

335337 

949 

989582 

47 

345755 

996 

654245 

30 

31 

9-335906 

948 

9-981)553 

47 

9-340353 

994 

10-653647 

29 

32 

336475 

946 

989525 

47 

346949 

9i)3 

653051 

28 

33 

337043 

945 

989497 

47 

347545 

992 

652455 

2? 

34 

337610 

944 

981)469 

47 

348141 

991 

651859 

26 

35 

338176 

943 

989441 

47 

348735 

91)0 

651265 

25 

36 

338742 

941 

989413 

47 

349329 

988 

650671 

24 

37 

339306 

940 

989384 

47 

349922 

987 

650078 

23 

38 

339871 

939 

989356 

47 

350514 

986 

649486 

22 

39 

340434 

937 

98!)3'28 

47 

351106 

985 

64881)4 

21 

40 

340996 

936 

989300 

47 

351697 

983 

648303 

20 

41 

9-341558 

935 

9-989271 

47 

9-352287 

982 

10647713 

19 

42 

342119 

934 

989243 

47 

352876 

981 

647124 

18 

43 

342679 

932 

989214 

47 

353465 

980 

646535 

17 

44 

343239 

931 

989186 

47 

354053 

979 

645947 

16 

45 

343797 

930 

989157 

47 

354640 

977 

645360 

15 

46 

344355 

929 

989128 

48 

355227 

976 

644773 

14 

47 

344912 

927 

989100 

48 

355813 

975 

644187 

13 

48 

345469 

926 

989071 

48 

356398 

974 

643602 

12 

49 

346024 

925 

989042 

48 

35(5982 

973 

643018 

11 

50 

346579 

924 

989014 

48 

357566 

971 

,642434 

10 

51 

9-347134 

922 

9-988985 

48 

9-358149 

970 

10-641851 

9 

52 

347(587 

921 

9HH956 

48 

358731 

969 

6412(19 

0 

53 

348240 

920 

988927 

48 

351(313 

968 

640687 

7 

54 

348792 

9J9 

988898 

48 

359893 

i>r,7 

040107 

6 

55 

349343 

917 

988869 

48 

360474 

966 

63952G 

5 

56 

349893 

916 

988H40 

48 

361053 

965 

638947 

4 

57 

350443 

915 

988811 

49 

36J63-J 

963 

638368 

3 

58 

350992 

914 

988782 

49 

3IW210 

962 

637700 

2 

59 

351540 

913 

988753 

49 

302787 

961 

637213 

1 

60 

352088 

911 

988724 

49 

363364 

960 

636636 

0 

Cosine      I 


I      Coung.     | 


Tang. 


77  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (13  Degrees.)      211 


|         Sine 


Cosine       I        D.      I       Tan<*. 


Cotang.       | 


0 

9-352088 

911 

9-988724 

49 

9-303364 

960   10-030636 

(K) 

1 

352035 

910 

988095 

49 

303940 

959 

030000 

59 

2 

353181 

909 

988060 

49 

364515 

958 

035485 

58 

3 

353720 

908 

988(536 

49 

365090 

957 

634910 

57 

4 

354271 

907 

988607 

49 

305004 

955 

(534336 

56 

5 

354815 

905 

988578 

49 

306237 

954 

633703 

55 

6 

355358 

904 

988548 

49 

360810 

953 

033190 

54 

7 

355901 

903 

988519 

49 

367382 

952 

632618 

53 

8 

356443 

902 

988489 

49 

367953 

951 

632047 

52 

9 

35(1984 

901 

9884(50 

49 

368524 

950 

631476 

51 

10 

357524 

899 

988430 

49 

3(59094 

949 

630906 

50 

11 

9-358064 

898 

9-988401 

49 

9-309663 

948 

10-630337 

49 

12 

358603 

897 

988371 

49 

370232 

946 

0297(58 

48 

13 

359141 

896 

988342 

49 

370799 

945 

629201 

47 

14 

359078 

895 

988312 

50 

371367 

944 

028033 

46 

15 

360215 

893 

988282 

50 

371933 

943 

(5281)67 

45 

16 

360752 

892 

988252 

50 

372499 

942 

627501 

44 

17 

361287 

891 

988223 

50 

373004 

941 

626936 

43 

18 

361822 

890 

988193 

50 

373629 

940 

620371 

42 

19 

362356 

889 

988163 

50 

374193 

939 

025807 

41 

20 

3(52889 

888 

988133 

50 

374756 

938 

625244 

40 

21 

9-363422 

881 

9-988103 

50 

9-375319 

937 

10-624681 

39 

22 

303954 

885 

988073 

50 

375881 

935 

624119 

38 

2:5 

364485 

884 

988043 

50 

376442 

934 

623558 

37 

24 

365016 

883 

988013 

50 

377003 

933 

622997 

36 

25 

365540 

882 

987983 

50 

377563 

932 

G22437 

35 

26 

306075 

881 

987953 

50 

378122 

931 

621878 

34 

27 

366604 

880 

987922 

50 

378081 

930 

621319 

33 

28 

367131 

879 

9878!)2 

50 

379239 

929 

620761 

32 

29 

367(559 

877 

987862 

50 

379797 

928 

620203 

31 

30 

368185 

876 

987832 

51 

38(1354 

927 

G  19646 

30 

31 

9-368711 

875 

9-987H01 

51 

9-3809  JO 

926 

10-019090 

29 

32 

369236 

874 

987771 

51 

381466 

925 

618534 

28 

33 

369701 

873 

987740 

51 

382020 

924 

617980 

27 

34 

370285 

872 

987710 

51 

382575 

923 

617425 

26 

35 

370808 

871 

987679 

51 

383129 

922 

610871 

25 

3(5 

371330 

870 

987649 

51 

:*83682 

921 

616318 

24 

37 

371852 

8(59 

987618 

51 

384234 

920 

615766 

23 

38 

372373 

807 

987588 

51 

384786 

919 

615214 

22 

3<J 

372894 

806 

987557 

51 

385337 

918 

614663 

21 

40 

373414  ' 

865 

987526 

51 

385888 

917 

614112 

20 

41 

9-373933 

864 

9-987496 

51 

9-386438 

915 

10-613562 

19 

42 

374452 

863 

987405 

51 

380987 

914 

613013 

18 

43 

374970 

862 

987434 

51 

387536 

913 

612404 

17 

44 

375487 

861 

987403 

52 

388084 

912 

611916 

16 

45 

37(5003 

800 

987372 

52 

388631 

911 

611309 

15 

41. 

376519 

859 

98734] 

52 

389178 

910 

610822 

14 

47 

377035 

858 

987310 

52 

389724 

909 

610276 

13 

48 

377549 

857 

987279 

52 

390270 

908 

609730 

12 

49 

378063 

856 

987*48 

52 

390815 

907 

609185 

11 

50 

378577 

854 

987217 

52 

391360 

906 

608640 

10 

51 

9-379089 

853 

9-987186 

52 

9-391903 

905 

10-608097 

9 

52 

379001 

852 

987155 

52 

392447 

904 

607553 

"  8 

53 

380113 

851 

987124 

52 

392989 

903 

607011 

7 

54 

380024 

850 

987092 

52 

393531 

902 

606469 

6 

55 

381134 

849 

987061 

52 

394073 

901 

605927 

5 

56 

381043 

848 

987030 

52 

394614 

900 

605386 

4 

57 

382152 

847 

980n98 

52 

395154 

899 

604846 

3 

58 

382061 

846 

98(5907 

52  • 

395(594 

898 

604300 

2 

59 

383168 

845 

986936 

52 

396233 

897 

603707 

60 

383675 

844 

986904 

52 

396771 

896 

603229 

0 

Cwme 


|       Sine 


|       Co;ang. 


T  a.,*. 


76  Decrees. 


212      (H  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |   Sine    |   D.   j   Cosine   1   D.  |   Tang.      D.      Cotanff.   | 

0  i  9-383675 

844 

9-980904 

52 

9-396771 

896 

10-OC3229 

1 

384182 

843 

980873 

53 

397309 

896 

CC2C91 

2 

384G87 

842 

986841 

53 

397846 

895 

6(,2154 

3 

38511)2 

841 

986809 

53 

398383 

'894 

601617 

4 

385097 

840 

980778 

53 

398919 

893 

601G81 

5 

386201 

839 

986746 

53 

399455 

892 

600545 

6 

38G704 

838 

986714 

53 

399990 

891 

60(1010 

7 

387207 

837 

980683 

53 

400524 

890 

599476 

8 

387709 

836 

9P6651 

53 

401058 

889 

598942 

9 

388210 

835 

986019 

53 

401591 

888 

598409 

10 

388711 

834 

986587 

53 

402124 

887 

597876 

n 

9-38921  1 

833 

9-986555 

53 

9-402656 

886 

10*597344 

12 

389711 

832 

980523 

53 

403187 

885 

59C813 

13 

390210 

831 

986-191 

53 

403718 

884 

596282 

14 

390708 

830 

980459 

53 

404249 

883 

595751 

15 

39J206 

828 

980-1-27 

53 

404778 

882 

595222 

16 

391703 

827 

986395 

53 

405308 

881 

594692 

17 

392199 

826 

986363 

54 

405836 

880 

594164 

18 

392695 

825 

986331 

54 

400364 

879 

593630 

19 

393191 

824 

986299 

54 

406892 

878 

593108 

20 

393085 

823 

986-266 

54 

407419 

877 

592581 

21 

9-394179 

822 

9-9P0234 

54 

9-407945 

876 

10-592055 

22 

394G73 

821 

986-202 

54 

408471 

875 

591529 

23 

395160 

820 

986169 

54 

408997 

874 

591003 

24 

395058 

819 

986137 

54 

409521 

874 

590479 

25 

390150 

818 

986104 

54 

410(145 

873 

589955 

26 

3UU641 

817 

980072 

54 

410569 

872 

589431 

27 

397  J  32 

817 

980039 

54 

41JI192 

871 

588908 

28 

397621 

816 

980007 

54 

411615 

870 

588385 

29 

398111 

815 

985974 

54 

412137 

869 

587803 

30 

398600 

814 

985942 

54 

412658 

868 

587342 

31 

S-399088 

813 

9-985909 

55 

9-413179 

867 

10-58(iP21 

32 

399575 

812 

985876 

55 

413699 

866 

58030  J 

33 

400062 

811 

985843 

55 

414-219 

865 

585781 

34 

400549 

810 

985811 

55 

414738 

864 

585262 

35 

401035 

809 

985778 

55 

415257 

804 

584743 

3(5 

401520 

808 

985745 

55 

415775 

803 

584225 

37 

402005 

807 

985712 

55 

416293 

862 

583707 

38 

402489 

806 

985679 

55 

4J0810 

861 

583190 

39 

402972 

805 

985646 

55 

417326 

860 

582074 

40 

403455 

804 

9850J3 

55 

417842 

859 

•  5821  ;  1-8 

41 

9-403938 

803 

9-985580 

55 

9-418358 

858 

10-58  1042 

42 

404420 

802 

91*5547 

55 

418873 

857 

581127 

43 

404901 

801 

985514 

55 

419387 

856 

580013 

44 

40,1:582 

800 

985480 

55 

419901 

855 

580099 

45 

405862 

799 

985447 

55 

420415 

855 

579585 

46 

400341 

798 

985414 

56 

420927 

854 

579073 

47 

406820 

797 

985380 

56 

421440 

853 

5785(50 

48 

407299 

796 

985347 

56 

421952 

852 

578048 

49 

407777 

795 

985314 

56 

422-163 

851 

577537 

50 

408254 

794 

985280 

56 

422974 

850 

577026 

51 

9-408731 

794 

9-985247 

50 

9-423484 

849 

10-576516 

52 

401)207 

793 

985213 

50 

4-23993 

848 

576007 

53 

409IJ82 

792 

985180 

56 

424503 

848 

575497 

54 

410157 

791 

985146 

56 

4-25011 

847 

574989 

55 

410032 

790 

9851  13 

56 

425519 

846 

574481 

50 

411106 

789 

985079 

56 

420027 

845 

573973 

57 

41  1579 

788 

985045 

56 

426534 

844 

573466 

58 

412052 

787 

985011 

56 

427041 

843 

572959 

59 

412524 

786 

984978 

56 

427547 

843 

572453 

60 

412996 

785 

984944 

56 

428052 

842 

571948 

Sine         i  |      Cotaug.       | 

75  Degrees. 


Tang.        I    M. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (15  Degrees.)      213 


M.    |        Sine         |       D.         |       Cosine 


D.      | 


Cotang. 


0 

9-41091)6 

785 

9-984944 

57 

9-428052 

842 

10-571948 

60 

1 

413467 

784 

984910 

57 

428557 

841 

571443 

59 

2 

413938 

783     984876 

57 

429062 

840 

570938 

58 

414-108 

783 

98-1842 

57 

429566 

839 

570434 

57 

A 

414H78 

782 

984808 

57 

430070 

838 

569930 

56 

415347 

781 

984774 

57 

430573 

838 

569427 

55 

(1 

415815 

780 

984740 

57 

431075 

837 

568925 

54 

7 

416283 

779 

984706 

57 

431577 

836 

568423 

53 

8 

416751 

778 

984672 

57 

432079 

835 

567921 

52 

9 

417217 

777 

984637 

•  57 

432580 

834 

567420 

51 

10 

417684 

776 

984603 

57 

433080 

833 

566920 

50 

11 

9-418150 

775 

9-984569 

57 

9-433580 

832 

10-506420 

49 

12 

418615 

774 

984535 

57 

434080 

832 

565920 

48 

13 

419079 

773 

984500 

57 

434579 

831 

565421 

47 

14 

419544 

773 

984466 

57 

435078 

830 

564922 

46 

15 

420007 

772 

984432 

58 

435576 

829 

564424 

45 

16 

420470 

771 

984397 

58 

436073 

828 

563927 

44 

17 

420933 

770 

984363 

58 

436570 

828 

563430 

43 

18 

421395 

769 

984.158 

58 

437067 

827 

562933 

42 

19 

421857 

768 

984294 

58 

437563  . 

826 

562437 

41 

20 

422318 

767 

984259 

58 

438059 

825 

561941 

40 

21 

9-422778 

767 

9-9S4224 

58 

9-438554 

824 

10-561446 

39 

22 

423238 

766 

984190 

58 

439048 

823 

560952 

38 

23 

423697 

765 

984155 

58 

439543 

823 

560457 

37 

24 

424156 

764 

984120 

58 

440036 

822 

559964 

36 

95 

424G15 

763 

984085 

58 

440529 

821 

559471 

35 

26 

425073 

762 

984050 

58 

441022 

820 

558978 

34 

27 

425530 

761 

984015 

58 

441514 

819 

558486 

33 

28 

425987 

760 

983981 

58 

442006 

819 

557994 

32 

29 

426443 

760 

983946 

58 

442497 

818 

557503 

31 

30 

426899 

759 

98391  1 

58 

442988 

817 

557012 

30 

31 

9-427354 

758 

9-983875 

58 

9-443479 

816 

10-556521 

29 

32 

427809 

757 

983840 

59 

443968 

'816 

556032 

28 

33 

428263 

756 

983805 

59 

444458 

815 

555542 

27 

34 

428717 

755 

983770 

59 

444947 

814 

555053 

26 

35 

429170 

754 

983735 

59 

445435 

813 

554565 

25 

36 

429623 

753 

983700 

59 

445923 

812 

554077 

24 

37 

430075 

752 

9836(54 

59 

446411 

812 

553589 

23 

38 

430527 

752 

983629 

59 

446898 

811 

553102 

22 

39 

430978 

751 

983594 

59 

447384 

810 

552616 

21 

40 

431429 

750 

983558 

59 

447870 

809 

552130 

20 

41 

J-431879 

749 

9-983523 

59 

9-448356 

809 

10-551644 

19 

42 

432329 

749 

983487 

59 

448541 

808 

551159 

18 

43 

432778 

748 

983452 

59 

449326 

807 

550674 

17 

44 

433226 

747 

983416 

59 

449810 

806 

550190 

16 

45 

433675 

746 

983381 

59 

450294 

806 

549706 

15 

46 

434122 

745 

983345 

59 

450777 

805 

549223 

14 

47 

434569 

744 

983309 

59 

451260 

804 

548740 

13 

48 

435016 

744 

983273 

60 

451743 

803 

548257 

12 

48 

435462 

743 

983238 

60 

452225 

802 

547775 

11 

50 

435908 

742 

983202 

60 

452706 

802 

547294 

10 

51 

9-436353 

741 

0-983166 

60 

9-453187 

801 

0-54(i813 

9 

52 

436798 

740 

983130 

60 

453668 

800 

546332 

8 

53 

437242 

740 

983094 

60 

454148 

799 

545852 

7 

54 

437686 

739 

983058 

60 

454628 

799 

545372 

6 

55 

438129 

738 

983022 

60 

455107 

798 

544893 

5 

56 

438572 

737 

982986 

60 

455586 

797 

544414 

4 

57 

439014 

736 

982950 

60 

4560(54 

796 

543936 

3 

58 

439456 

736 

982914 

60 

456542 

796 

543458 

2 

59 

439897 

735 

982878 

60 

457019 

795 

542981 

1 

60 

440338 

734 

982842 

60 

457496 

794 

542504 

0 

Sine        I  |     Coiang. 

74  Degrees. 


I      Tang. 


214      (16  Degrees.)    LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Sine    |   D.   |   Cosine   |   D.      Tang.      D.   |   Cofan£.  | 

0 

9-440338 

734 

9-982642 

60 

9-4574%    794 

10-542504 

•  ill 

1 

440778 

733 

982805 

00 

457973 

.793 

542027 

59 

2 

441218 

732 

982769 

01 

458449 

793 

541551 

58 

3 

441058 

731 

982733 

01 

4581)25 

792 

541075 

57 

4 

442096 

731 

98209G 

01 

459400 

791 

540000 

r><; 

5 

44-2535 

730 

98-2000 

01 

459875 

7i>0 

540125 

55 

6 

442973 

729 

982624 

01 

400349 

790 

539051 

51 

7 

443410 

728 

982587 

61 

400823 

7fJ9 

539177 

53 

8 

443847 

727 

983551 

01 

401297 

788 

538703 

52 

9 

4-14284 

727 

98-2514 

01 

401770 

788 

538230 

51 

10 

444720 

720 

98-2477 

61 

462342 

787 

537758 

50 

11 

9-445155 

725 

9-982441 

61 

9-402714 

780 

10-537286 

49 

12 

445590 

724 

982404 

01 

403180 

785 

530814 

48 

13 

44G025 

723 

982307 

01 

403058 

785 

530342 

47 

14 

446459 

723 

982331 

61 

404120 

784 

535871 

40 

15 

446893 

722 

982294 

61 

464599 

783 

535401 

45 

1G 

447320 

721 

982257 

61 

405009 

783 

534931 

44 

17 

447759 

720 

982220 

02 

405539 

782 

534401 

43 

18 

448191 

720 

982183 

02 

400008 

781 

533992 

12 

19 

448023 

719 

982140 

62 

400470 

780 

533524 

41 

20 

449054 

718 

982109 

62 

400945 

780 

533055 

40 

21 

9-449485 

717 

9-982072 

62 

9-407413 

779 

/0-532587 

39 

22 

449915 

710 

982035 

62 

407880 

778 

532120 

38 

23 

450345 

716 

981993 

62 

408347 

778 

531053 

37 

24 

450775 

715 

981901 

62 

4G8814 

777 

531180 

30 

25 

451204 

714 

981924 

62 

409280 

770 

530720 

35 

26 

451032 

713 

981880 

62 

409740 

775 

530254 

34 

27 

452000 

713 

981849 

62 

470211 

775 

529789 

33 

28 

452488 

"712 

981812 

62 

470G7G 

774 

529324 

32 

29 

452915 

711 

981774 

62 

471141 

773 

528859 

31 

30 

453342 

710 

981737 

62 

471005 

773 

528395 

30 

31 

9-453708 

710 

9-981099 

63 

9-472008 

772 

10-527932 

29 

32 

454194 

709 

981002 

63 

472532 

771 

527408 

28 

33 

454019 

708 

981025 

63 

4721);!.-) 

771 

527005 

27 

34 

455044 

707 

981587 

63 

473457 

770 

520543 

20 

35 

455409 

707 

981549 

63 

473919 

709 

520081 

25 

36 

455893 

700 

981512 

63 

474381 

7(59 

525019 

24 

37 

450310 

705 

981474 

63 

474842 

708 

525158 

23 

38 

450739 

704 

981430 

63 

475303 

707 

524097 

22 

39 

457102 

704 

981399 

63 

475703 

707 

524237 

21 

40 

457584 

703 

981301 

63 

470223 

700 

523777 

20 

41 

9-458006 

702 

9-981323 

63 

9-470083 

783 

10-523317 

19 

42 

458427 

701 

981235 

63 

477142 

7(55 

522858 

18 

43 

458848 

701 

981247 

63 

477001 

704 

522399 

17 

44 

459208 

700 

981209 

63 

478059 

703 

521941 

10 

45 

459(588 

699 

981171 

63 

478517 

7(53 

521483 

15 

46 

400108 

698 

981133 

64 

478975 

702 

521025 

14 

47 

400527 

698 

981095 

64 

479432 

701 

520508 

13 

48 

400940 

697 

981057 

64 

479889 

701 

520111 

12 

49 

401304 

690 

981019 

64 

480345 

700 

519055 

11 

50 

401782 

095 

980981 

64 

480801 

759 

519199 

10 

51 

9-402199 

695 

9-980942 

64 

9-481257 

759 

10-518743 

9 

52 

402010 

094 

980904 

64 

481712 

758 

5J8288 

8 

53 

403032 

693 

9808(56 

64 

482(07 

757 

517833 

7 

54 

403448 

093 

980827 

64 

482021 

757 

517379 

0 

55 

4G3804 

092 

980789 

64 

483075 

750 

510925 

5 

56 

404279 

091 

980750 

64 

483529 

755 

510471 

4 

57 

404094 

090 

980712 

04 

483982 

755 

510018 

3 

58 

405108 

090 

980073 

04 

48441(5 

754 

515505 

2 

59 

405522 

089 

980035 

04 

484887 

753 

515113 

1 

60 

405935 

688 

980590 

64 

485339 

753 

514001 

0 

I     Cosii 


1      Cotang. 


Tang. 


73  Degrees 


LOGARITHMIC  SINES,  COSINES,  ETC.    (17  Degrees.)      215 


J*.  /  __Sme          |D.       |       Cosine        |      D.      |       Tang.        |       D. 


0 

9-405935 

688 

9-9S<)5% 

64 

9-4*5339 

755 

0-5H661 

M) 

1 

406348 

688 

980558 

64 

485791 

752 

514209 

r)<J 

2 

400761 

687 

9805  19 

65 

486242 

751 

513758 

r>8 

3 

467173 

686 

980480 

65 

48GC93 

751 

513307 

->7 

4 

467585 

685 

980442 

65 

487143 

750 

512857 

r>l> 

5 

407996 

685 

980403 

65 

487593 

749 

512407 

r>5 

r> 

468407 

684 

980304 

65 

488043 

749 

511957 

54 

408817 

683 

980325 

65 

488402 

748 

511508 

53 

8 

4119227 

683 

9802S6 

65 

488941 

747 

511059 

52 

9 

469637 

682 

980247 

65 

489390 

747 

5  ICG  10 

r>l 

10 

47004G 

C81 

986208 

65 

489838 

746 

510162 

50 

11 

9-470455 

680 

9-980169 

65 

9-490286 

746 

10-51,9714 

41) 

1-2 

470803 

680 

980130 

65 

490733 

745 

509267 

48 

13 

471271 

679 

980091 

65 

491180 

744 

51*8820 

47 

14 

471G79 

678 

980052 

65 

491627 

744 

508373 

46 

15 

472086 

678 

980012 

65 

492073 

743 

507927 

45 

16 

472492 

677 

979973 

65 

492519 

743 

507481 

44 

1? 

472898 

67G 

979934 

66 

4929G5 

742 

507035 

43 

18 

473304 

676 

979895 

66 

493410 

741 

506590 

42 

19 

4737  JO 

675 

979855 

66 

49:3854 

740 

506146 

41 

20 

474115 

674 

979816 

66 

494299 

740 

505701 

4!) 

21 

9-474519 

674 

9-979776 

66 

9-494743 

740 

10-505257 

3!l 

22 

474923 

673 

979737 

66 

495186 

739 

504814 

38 

23 

475327 

672 

979697 

66 

495630 

738 

504370 

37 

24 

475730 

672 

979058 

66 

496073 

737 

5(i3927 

36 

25 

47G133 

671 

979GI8 

C6 

49G515 

737 

503485 

35 

26 

476536 

670 

979579 

66 

496957 

736 

503043 

34 

27 

470938 

669 

979539 

66 

497399 

736 

502601 

33 

28 

477340 

6(19 

979499 

66 

497841 

735 

502159 

32 

29 

477741 

668 

979459 

66 

498282 

734 

501718 

31 

30 

478142 

6G7 

979420 

66 

498722 

734 

501278 

30 

3J 

9-478542 

667 

9-979380 

66 

9-499163 

733 

10-5CC837 

29 

32 

478942 

666 

979340 

GG 

499CC3 

733 

500397 

28 

33 

479:^42 

6(55 

979300 

67 

50CC42 

732 

499958 

27 

34 

479741 

6(15 

9792HO 

67 

500481 

731 

499519 

26 

35 

480140 

604 

979220 

67 

50QS20 

731 

499(J80 

25 

36 

480539 

663 

979180 

67 

601  359 

730 

498041 

24 

37 

480937 

663 

979140 

67 

501797 

730 

498203 

23 

38 

48  J  334 

662 

979100 

C7 

502235 

729 

4977(15 

22 

3D 

481231 

661 

979059 

67 

5C2072 

728 

497328 

21 

40 

482128 

661 

979019 

67 

503109 

728 

496891 

2(J 

41 

9-48252.') 

GOO 

9-978979 

67 

9-503546 

727 

10-496454 

19 

42 

482921 

659 

978939 

67 

503982 

727 

49G018 

Id 

43 

483316 

659 

978898 

67 

504418 

726 

495582 

17 

44 

483712 

658 

978£58 

67 

504854 

725 

495146 

If 

45 

484107 

657 

978,817 

67 

505289 

725 

494711 

i" 

46 

484501 

657 

978777 

67 

505724 

724 

494276 

n 

47 

484895 

656 

978736 

67 

500159 

724 

49.'i841 

i: 

48 

485289 

655 

978696 

68 

506593 

723 

493407 

IS 

49 

485682 

655 

978G55 

68 

507027 

722 

492973 

11 

50 

48G075 

654 

978615 

68 

507460 

722 

492540 

11 

51 

9-480407 

653 

9-978574 

68 

9-507893 

721 

10-492107 

< 

52 

4815800 

653 

978533 

68 

508326 

721 

491674 

1 

53 

487251 

652 

978493 

68 

508759 

720 

491241 

54 

487(i43 

651 

978452 

68 

509191 

719 

49G809 

< 

55 

488034 

651 

978411 

68 

509622 

719 

490378 

5(i 

488424 

650 

978370 

68 

510054 

718 

489946 

, 

57 

48«814 

650 

97S529 

68 

510485 

718 

489515 

58 

489204 

649 

978288 

68 

510916 

717 

4891  !84 

59 

489593 

648 

978247 

68 

511340 

716 

488654 

60 

489982 

648 

9782C6 

68 

511776 

716 

488224 

I     .Cosme       | 


I       Colang.       | 


I         Ta,,S. 


72  Degrees 


216      (18  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


I      P.      I        Tang.         |        D.'     |       Cn 


(1 

9*489982 

648 

9-978206 

68 

9-51  1776 

716 

10-48*224 

00 

1 

490371 

648 

978165 

68 

512206 

716 

487794 

o9 

2 

490759 

647 

978124 

68 

512635 

715 

4873(i5 

58 

3 

491147 

646 

978083 

69 

5130(34 

714 

486936 

r,7 

4 

491535 

646 

978042 

69 

513493 

714 

486507 

M 

5 

491922 

645 

978001 

69 

513921 

713 

486079 

i>:> 

6 

492308 

644 

977JJ59 

69 

514349 

713 

485051 

54 

7 

492G95 

644 

977918 

69 

514777 

712 

485223 

53 

8 

493081 

643 

977877 

69 

515-204 

712 

48471)6 

52 

9 

493466 

642 

977835 

69 

515631 

711 

484309 

:.i 

10 

493851 

642 

977794 

69 

51G057 

710 

483943 

50 

11 

9-494236 

641 

9-977752 

69 

9-510484 

710 

10-483516 

49 

IS 

494021 

641 

977711 

69 

510910 

709 

483090 

48 

13 

495005 

640 

977669 

69 

517335 

709 

482005 

47 

14 

495388 

639 

977G28 

69 

517761 

708 

482239 

4(i 

15 

495772 

639 

977586 

69 

518185 

708 

481815 

45 

10 

496154 

638 

977544 

70 

518610 

707 

481390 

44 

17 

496537 

637 

977503 

70 

519034 

706 

480966 

43 

18 

49G919 

637 

9774(51 

70 

519458 

706 

480542 

4-2 

19 

497301 

636 

977419 

70 

519882 

705 

480118 

41 

20 

497682 

636 

977377 

70 

520305 

705 

479095 

40 

21 

9-498064 

635 

9-977335 

70 

9-520728 

704 

10-479272 

39 

22 

498444 

634 

977293 

70 

521151 

703 

478849 

38 

23 

498825 

634 

977251 

70 

521573 

703 

478427 

37 

24 

499204 

633 

977209 

70 

521995 

703 

478005 

3(5 

25 

499584 

632 

977167 

70 

522417 

702 

477583 

35 

2(i 

499963 

632 

977125 

70 

5228:18 

702 

477162 

34 

27 

500342 

631 

977083 

70 

523259 

701 

476741 

33 

28 

500721 

631 

977041 

70 

523680 

701 

470320 

32 

29 

501099 

630 

970999 

70 

524100 

700 

475900 

31 

30 

501476 

629 

970957 

70 

524520 

699 

475480 

30 

31 

9-50  1H54 

G29 

9-97f914 

70 

9-524939 

699 

10-475001 

29 

3-2 

502231 

628 

970872 

71 

525359 

698 

474041 

28 

33 

502<H)7 

628 

1)70830 

71 

525778 

698 

4742-22 

27 

34 

502-184 

627 

970787 

71 

5%  197 

697 

473803 

'26 

35 

503360 

626 

976745 

71 

526615 

697 

473385 

25 

36 

503735 

626 

976762 

71 

5-27033 

696 

472967 

•24 

37 

504110 

625 

9766*10 

71 

527451 

696 

47-2549 

23 

38 

504485 

6-25 

97C617 

71 

527868 

695 

472132 

2-2 

3D 

504860 

624 

976574 

71 

528285 

695 

471715 

21 

40 

505234 

623 

976532 

71 

528702 

694 

471-298 

20 

41 

9-505008 

623 

9-976489 

71 

9-529119 

693 

10-470881 

19 

42 

505981 

622 

97<>446 

71 

529535 

693 

470405 

18 

43 

506354 

622 

970404 

71 

52J950 

693 

470050 

17 

44 

506727 

621 

970301 

71 

530306 

692 

469034 

JO 

45 

507099 

620 

976318 

71 

530781 

691 

409219 

J5 

46 

507471 

620 

976275 

71 

531196 

691 

468804 

11 

47 

507H43 

619 

97G232 

72 

531611 

690 

468389 

13 

48 

508214 

619 

976189 

72 

532025 

690 

407975 

1-2 

49 

508585 

618 

976146 

72 

53-2439 

689 

467561 

11 

50 

508956 

618 

976103 

72 

532853 

689 

407147 

10 

51 

9-509326 

617 

9-976060 

72 

9-533266 

688 

10-406734 

9 

52 

509696 

616 

976017 

72 

533079 

688 

460321 

8 

53 

510065 

616 

975974 

72 

5340U2 

687 

405908 

7 

54 

510434 

615 

975930 

72 

534504 

687 

465496 

0 

55 

510803 

615 

975887 

72 

534916 

686 

405084 

5 

5fi 

511172 

614 

975844 

72 

535328 

686 

404672 

4 

57 

511540 

613 

975800 

72 

535739 

685 

404-261 

3 

58 

511907 

613 

975757 

72 

530150 

685 

403850 

2 

59 

512275 

612 

975714 

72 

536561 

684 

403439 

GO 

512642 

612 

975670 

72 

536972 

684 

463028 

0 

Colang. 


I         Tang. 


U  Decrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (19  Degrees.)      217 


Sine    !   D.   |   Cosine      D.      Tulip;.      D.   \   Cotan?. 

9-  -.12642 

612 

9-975670 

73 

9-336972    684 

10-463028 

60 

513009 

611 

975(527 

73 

537382 

(583 

46.2618 

59 

513375 

611 

975583 

73 

5377!  12 

183 

462208 

58 

513741 

610 

975539 

73 

538902 

682 

461798 

57 

514107 

609 

975496 

73 

5386  1J 

682 

4(51389 

50 

514472 

609 

975452 

73 

539020 

681 

4(50980 

55 

514837 

608 

975408 

73 

539429 

681 

4(50571 

54 

515302 

608 

975°G5 

73 

539P37 

680 

460163 

53 

515506 

607 

9/5321 

73 

54(3245 

680 

459755 

52 

515930 

607 

975277 

73 

540G53 

679 

459347 

51 

516294 

606 

975233 

73 

541061 

679 

458939 

50 

9-516657 

605 

9-975189 

73 

9-541468 

678 

10-458532 

49 

517020 

605 

975145 

73 

541875 

678 

458125 

48 

517382 

604 

975101 

73 

542281 

677 

457719 

47 

517745 

604 

975057 

73 

542G88 

677 

457312 

46 

518107 

603 

975013 

73 

543094 

676 

456906 

45 

5184G8 

603 

974969 

74 

543499 

676 

456501 

44 

518829 

602 

974925 

74 

543905 

675 

456095 

43 

519190 

601 

974880 

74 

544310 

675 

455690 

42 

519551 

601 

974836 

74 

544715 

674 

455285 

41 

519911 

600 

9747J2 

74 

545119 

674 

454881 

40 

9-533271 

600 

9-974748 

74 

9-545524 

673 

10-454476 

39 

520G31 

599 

974703 

74 

545928 

673 

454072 

33 

520990 

599 

974659 

74 

54G331 

672 

453(569 

37 

521349 

598 

974614 

74 

54G735 

672 

453265 

36 

521707 

593 

974570 

74 

547138 

671 

432862 

35 

52206G 

597 

974523 

74 

547540 

671 

4524(50 

34 

522424 

596 

974481 

74 

547943 

670 

452057 

33 

522781 

596 

974436 

74 

548345 

670 

451655 

32 

523,138 

595 

974391 

74 

548747 

669 

431253 

31 

523495 

595 

974347 

75 

549149 

669 

450851 

30 

9-523852 

594 

9-974302 

75 

9-543550 

668 

10-450450 

29 

524308 

594 

974257 

75 

54-W51 

6G8 

45i)049 

28 

524504 

593 

974212 

75 

550352 

667 

44JG48 

27 

524920 

593 

974  J  67 

75 

550752 

667 

449248 

26 

525275 

5U2 

974122 

75 

551152 

666 

448848 

25 

525G30 

591 

974077 

75 

551552 

666 

448448 

24 

525984 

591 

974032 

75 

551932 

665 

448048 

23 

52G339 

590 

973987 

75 

552351 

665 

447649 

22 

52GG93 

590 

973942 

75 

552750 

665 

447250 

21 

527046 

589 

973897 

75 

553149 

664 

446851 

20 

9-527400 

589 

9-973852 

75 

9-553548 

664 

10-446452 

19 

527753 

588 

973807 

75 

553946 

663 

446054 

18 

528105 

588 

973761 

75 

554344 

663 

445656 

17 

528458 

587 

973716 

76 

554741 

6(52 

445259 

16 

528810 

587 

973671 

76 

555139 

662 

444861 

15 

52'J161 

586 

973625 

76 

555.536 

661 

444464 

14 

529513 

586 

973580 

76 

553933 

661 

4440(57 

13 

529864 

585 

973535 

76 

55(5329 

660 

443671 

12 

530215 

585 

973489 

76 

536725 

660 

443275 

11 

530565 

584 

973444 

76 

557121 

659 

442879 

10 

9530915 

584 

9-973398 

76 

9-557517 

659 

10-442483 

9 

531265 

583 

973352 

76 

557913 

659 

442087 

8 

531614 

582 

973307 

76 

558308 

658 

441692 

7 

531963 

582 

973261 

76 

558702 

658 

441298 

6 

532312 

581 

973215 

76 

559097 

657 

440903 

5 

532661 

581 

9731(59 

76 

539491 

657 

440509 

4 

533009 

580 

973124 

76 

559885 

656 

440J  15 

3 

533357 

580 

973078 

76 

560279 

656 

439721 

2 

533704 

579 

973032 

77 

560673 

655 

439327 

] 

.53-1052 

578 

972986 

77 

561066 

655 

438934 

0 

|      Cotiae 


|      Cotang.     I 


1        Tang.       |    M. 


70  Dtgreea. 


218      (20  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


Sine         |        D.       |       Cosine        |      D.      |       Tan<r.        |       D.       |       Cotanjr.      | 


0 

9-534052 

578 

9-ii7-2!is(5 

77 

9-5(5  1066 

655 

10-43.S034 

(51) 

] 

534399 

577 

972.140 

77 

56  J  459 

654 

43S54I 

59 

2 

534745 

577 

972MJ4 

77 

5(51851 

654 

438149 

te 

3 

535092 

577 

972848 

77 

562244 

653 

4377,16 

f>7 

4 

535-1:18 

576 

972802 

77 

562636 

653 

4373(54 

56 

5 

535783 

576 

972755 

77 

563028 

653 

430972 

53 

6 

53(5129 

575 

972709 

77 

563419 

652 

43(5581 

54 

7 

536474 

574 

972063 

77 

563811 

652 

43(5189 

53 

8 

53(3818 

574 

972617 

77 

5(54-202 

651 

435798 

53 

9 

5371G3 

573 

972570 

77 

564  .V.  12 

651 

435408 

r>l 

in 

537507 

573 

972524 

77 

564983 

650 

43.~>017 

50 

11 

9-537851 

572 

9-972478 

77 

9-565373 

650 

10-4346-27 

49 

12 

538194 

572 

97-2431 

78 

5657(53 

649 

434-237 

48 

13 

538538 

571 

972385 

78 

5(56153 

(549 

433847 

47 

14 

5388^0 

571 

972338 

78 

560542 

649 

433458 

46 

15 

539-223 

570 

972291 

78 

566932 

648 

433068 

45 

16 

539565 

570 

972-245 

78 

567320 

648 

432680 

44 

17 

539907 

569 

972198 

78 

567709 

647 

432291 

43 

18 

540249 

569 

972151 

78 

568098 

(547 

431902 

4-2 

19 

540590 

508 

972105 

78 

568486 

646 

431514 

41 

20 

540931 

568 

972058 

78 

568873 

646 

431127 

40 

21 

9-54  J272 

567 

9-972011 

78 

9-5692(51 

645 

10-430739 

39 

22 

541613 

567 

971964 

78 

5(51)648 

645 

430352 

38 

23 

541953 

566 

971917 

78 

570035 

645 

4-2911(55 

37 

24 

542-21I3 

566 

971870 

78 

570422 

644 

429578 

3(3 

25 

54263-2 

565 

971823 

78 

570809 

(544 

421)191 

35 

26 

542!»71 

565 

971776 

78 

571  11)5 

643 

428805 

34 

27 

543310 

564 

971729 

79 

571581 

643 

428419 

33 

28 

543649 

564 

971682 

79 

5711)67 

642 

428033 

3-3 

29 

543987 

563 

971635 

79 

572352 

(542 

427648 

31 

30 

544325 

563 

971588 

79 

572738 

642 

427262 

30 

31 

9-544663 

562 

9-971f>40 

79 

9-573123 

641 

10-42(5877 

•2!) 

32 

545000 

562 

971493 

79 

573.-)07 

641 

4-264113 

•28 

33 

545338 

561 

971446 

79 

573892 

640 

426108 

27 

34 

545(574 

561 

971398 

79 

574276 

640 

425724 

-26 

35 

546011 

560 

971351 

79 

574660 

639 

42:>34() 

25 

36 

54(i347 

560 

971303 

79 

575044 

639 

424956 

-24 

37 

546683 

559 

971256 

79 

575427 

639 

4-24:>73 

-23 

38 

547019 

559 

971208 

79 

575810 

638 

424190 

-2-2 

39 

547354 

558 

971161 

79 

576193 

638 

4-23H07 

21 

40 

547689 

558 

971113 

79 

576576 

637 

423424 

20 

41 

9-548024 

557 

9-971066 

80 

9-576958 

637 

10-423041 

IS) 

42 

548359 

557 

971018 

80 

577341 

638 

422659 

18 

43 

548693 

556 

970970 

80 

5777-23 

636 

422277 

17 

44 

549027 

556 

970922 

80 

578104 

636 

421896 

16 

45 

541)260 

555 

970H74 

80 

578486 

635 

421514 

15 

46 

549093 

555 

970827 

80 

578867 

(535 

421133 

14 

47 

550026 

554 

970779 

80 

579248 

e:J4 

420752 

13 

48 

550359 

5.54 

"70731 

80 

579629 

634 

42U371 

1-2 

49 

550692 

553 

970(583 

80 

580009 

634 

419991 

11 

50 

551024 

553 

970635 

80 

580389 

633 

419611 

10 

51 

9-551356 

552 

9-970~>86 

80 

9-580769 

633 

10-419231 

9 

52 

551687 

552 

9705:58 

80 

581149 

632 

41  HH51 

8 

53 

552018 

552 

9704!M) 

80 

581528 

632 

418472 

7 

54 

552349 

551 

97044-2 

80 

581907 

632 

4181)93 

6 

55 

552680 

551 

970394 

80 

582286 

631 

4I77I4 

5 

56 

553010 

550 

970345 

81 

582(565 

631 

41733$ 

4 

57 

553341 

550 

970297 

81 

583043 

630 

41(5957 

3 

58 

553070 

549 

'.(70-249 

81 

5H34-22 

cno 

41C>578 

I 

59 

554000 

549 

9702IX) 

81 

583800 

629 

410200 

1 

60 

554329 

548 

970152 

81 

584177 

629 

415823 

0 

|       Cosine        | 


|         Sine         | 


Couuig.      1 


Tang.       | 


Dejreea 


LOGARITHMIC  SINES,  COSINES,  ETC.    (21  Degrees.)      219 


M.  |    Sine       D.   |    Cosine   |   D.    ~  Tan?.      D.       Cotnng. 

0 

9-554329 

543 

9-970152  1  81 

9-584177 

G29 

10-415823 

60 

554!  558 

548 

970103    81 

584555 

629 

415445 

59 

2 

554987 

547 

9"0055 

81 

584932 

628 

415008 

58 

3 

555315 

547 

970006 

81 

585309 

628 

414(591 

57 

4 

555643 

546 

9(59957 

81 

58508G 

627 

414314 

56 

5 

555971 

546 

960309 

81 

580002 

627' 

413938 

55 

6 

556299 

545 

9(59860 

81 

58043D 

627 

412561 

54 

7 

556(526 

545 

969811 

81 

586815 

G2G 

413185 

53 

8 

55(i953 

544 

9G9762 

81 

587190 

626 

41^310 

52 

9 

557-280 

544 

969714 

81 

587506 

625 

412434 

51 

10 

557606 

543 

909G65 

81 

587941 

625 

412059 

50 

!1 

9  557932 

543 

9-969616 

82 

9-588316 

625 

10-411084 

49 

12 

558358 

543 

9695(57 

82 

588691 

624 

411209 

48 

13 

558583 

542 

969518 

82 

5B90G6 

624 

41C934 

17 

14 

558909 

542 

969469 

82 

589-140 

623 

41C560 

46 

15 

559-234 

541 

969420 

82 

589814 

623 

410186 

45 

16 

559558 

541 

909370 

82 

5901  88 

623 

4C9812 

44 

17 

559883 

540 

969321 

82 

590562 

622 

409438 

43 

18 

560207 

540 

969272 

82 

590935 

622 

409065 

42 

1<J 

5(50531 

539 

969223 

82 

591308 

622 

4C8G92 

41 

20 

560855 

539 

969173 

82 

591081 

621 

408319 

40 

21 

9-561  178 

538 

9-909124 

82 

9-592054 

621 

10-407946 

39 

22 

561501 

538 

969075 

82 

592426 

620 

407574 

38 

2:5 

561824 

537 

969025 

82 

592798 

620 

407202 

37 

24 

562146 

537 

908976 

82 

593170 

619 

400829 

36 

2f> 

562468 

536 

968926 

83 

593542 

619 

400458 

35 

2ti 

562790 

536 

908877 

83 

593914 

618 

400086 

34 

27 

563112 

536 

908827 

83 

594285 

618 

405715 

33 

88 

563433 

535 

908777 

83 

594656  - 

618 

405344 

32 

39 

5(53755 

535 

968728 

83 

595027 

617 

404973 

31 

30 

5(54075 

534 

968678 

83 

595398 

617 

4046C2 

30 

31 

9-5(54396 

534 

9-968628 

83 

9-595768 

617 

10-404232 

29 

3-2 

5(54716 

533 

908578 

83 

590138 

616 

403802 

28 

33 

5(55036 

533 

908528 

83 

590508 

616 

403492 

27 

34 

565356 

532 

968479 

83 

590878 

616 

403122 

26 

35 

5(55676 

532 

9<!8429 

83 

597247 

615 

402753 

25 

3fi 

5(55995 

531 

9!;8379 

83 

597616 

615 

402384 

24 

37 

566314 

531 

908329 

83 

597985 

615 

402015 

23 

38 

56(5632 

531 

908278 

83 

598354 

614 

401646 

22 

39 

566951 

530 

968228 

84 

598722 

614 

401278 

21 

40 

567269 

530 

968178 

84 

59D091 

613 

400909 

20 

41 

9-567587 

529 

9-968128 

84 

9-599459 

613 

10-400541 

19 

42 

567904 

529 

968078 

84 

599827 

613 

400173 

18 

43 

568222 

528 

9G8027 

84 

600194 

612 

399806 

17 

44 

5(58539 

528 

967977 

84 

600502 

612 

399438 

16 

45 

568856 

528 

967927 

84 

600929 

611 

399071 

15 

40 

5(59172 

527 

96787G 

84 

601236 

611 

398704 

14 

47 

569488 

527 

967828 

84 

601C62 

611 

398338 

13 

48 

569804 

526 

907775 

84 

6(12029 

610 

397971 

12 

49 

570120 

526 

967725 

84 

602395 

610 

397605 

11 

50 

570435 

525 

907674 

84 

6C27G1 

610 

397239 

10 

51 

9-570751 

525 

9-907024 

84 

9-603127 

609 

10-290873 

9 

52 

5710G6 

524 

967573 

84 

603493 

G09 

3;!G5';7 

8 

S3 

571380 

524 

967522 

85 

C03858 

609 

SCO  142 

7 

54 

571G95 

523 

967471 

85 

604223 

608 

395777 

6 

55 

572009 

523 

967421 

85 

604588 

G08 

395412 

5 

50 

572323 

523 

967370 

85 

604953 

607 

31S5047 

4 

57 

572;-36 

522 

907319 

85 

'605317 

607 

394083 

3 

58 

572950 

522 

907268 

85 

605082 

607 

394318 

2 

SO 

573203 

521 

907217 

85 

606046 

606 

393954 

1 

60 

573575 

521 

907160 

85 

600410 

606 

393590 

0 

|       Cosini 


|       Cotang.      | 


I        Tang.        I  M. 


Degrees. 


220      (22  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Sine    |   D.       Cosine      D.   |   Tang1.       D.      Cotang-.   | 

0 

9-573575 

521 

9-967166 

85 

9-606410 

606 

10393590 

6(1 

1 

573383 

520 

967115 

85 

606773 

606 

393227 

59 

2 

574200 

520 

967064 

85 

607137 

605 

392863 

58 

3 

574512 

519 

967013 

85 

607500 

605 

392500 

57 

4 

574824 

519 

966961 

85 

607863 

604 

392137 

56 

5 

575136 

519 

9G6910 

85 

608225 

604 

391775 

55 

fi 

575447 

518 

966859 

85 

608588 

604 

391412 

54 

7 

575758 

518 

966808 

85 

608950 

603 

391050 

53 

8 

57GOG9 

517 

966756 

86 

609312 

603 

390688 

52 

9 

576379 

517 

966705 

86 

609674 

603 

390326 

51 

]() 

576689 

516 

966653 

86 

610036 

602 

389964 

50 

11 

9-576999 

516 

9-966632 

86 

9-610397 

602 

10-389603 

49 

12 

577309 

516 

96(5550 

86 

610759 

602 

389241 

48 

13 

577018 

515 

960499 

86 

61  1  120 

601 

388880 

47 

14 

577927 

515 

96(5447 

86 

611480 

601 

388520 

46 

15 

578236 

514 

066395 

86 

611841 

601 

388159 

45 

16 

578545 

514 

966344 

86 

612201 

600 

387799 

44 

17 

578853 

513 

986292 

86 

612561 

600 

387439 

43 

18 

579162 

513 

966240 

86 

612921 

600 

387079 

42 

19 

579470 

513 

966188 

86 

613281 

599 

386719 

41 

20 

579777 

512 

966136 

86 

613641 

599 

386359 

40 

21 

9-580085 

512 

9  966085 

87 

9-614000 

598 

10-386000 

39 

22 

580392 

511 

966033 

87 

614359 

598 

385641 

38 

2:< 

580699 

511 

965981 

87 

614718 

598 

385282 

37 

24 

581005 

511 

965928 

87 

615077 

597 

384923 

36 

25 

58KU  2 

510 

965876 

87 

615435 

597 

384565 

35 

36 

581618 

510 

965824 

87 

615793 

597 

384207 

34 

27 

581924 

509 

9G5772 

87 

61(5151 

596 

383849 

33 

28 

582229 

509 

965720 

87 

G16509 

596 

383491 

32 

29 

582535 

509 

9656(58 

87 

61G867 

596 

383133 

31 

3!J 

582840 

508 

965615 

87 

617224 

595 

382776 

30 

31 

•J-583145 

508 

9-965563 

87 

9-617582 

595 

10-382418 

29 

32 

583449 

507 

965511 

87 

617939 

595 

382061 

28 

33 

583754 

507 

965458 

87 

618295 

594 

381705 

27 

34 

584058 

506 

9(55406 

87% 

618652 

594 

381348 

2(5 

35 

584361 

506 

965353 

88 

619008 

594 

380992 

25 

30 

584665 

506 

9(55301 

88 

619364 

593 

380636 

24 

37 

5849G8 

505 

965248 

88 

619721 

593 

380279 

23 

38 

585272 

505 

965195 

88 

620076 

593 

379924 

22 

39 

585574 

504 

965143 

88 

620432 

592 

3795G8 

21 

40 

585877 

504 

965090 

88 

620787 

592 

379213 

20 

41 

9-586179 

503 

9-965037 

88 

9-621142 

592 

10-378858 

19 

42 

586482 

503 

9(54984 

88 

621497 

591 

378503 

18 

43 

586783 

503 

9<>4931 

88 

621852 

591 

378148 

17 

44 

587085 

502 

964879 

88 

622207 

590 

377793 

16 

45 

587386 

502 

964826 

88 

622561 

590 

377439 

15 

40 

587688 

501 

9(54773 

88 

622915 

590 

377085 

14 

47 

587989 

501 

964719 

88 

623269 

589 

376731 

13 

48 

588289 

501 

964060 

89 

623(523 

589 

376377 

12 

49 

588.390 

500 

964613 

89 

623976 

589 

376024 

11 

50 

588890 

500 

964560 

89 

624330 

588 

375670 

10 

51 

9-589190 

499 

9-964507 

89 

9-624683 

588 

10-375317 

9 

52 

589489 

499 

964454 

89 

(525036 

588 

374964 

8 

53 

5S97H9 

499 

964400 

89 

625388 

587 

374612 

7 

54 

590088 

498 

964347 

89 

625741 

587 

374259 

6 

53 

590387 

498 

964294 

89 

62(5(193 

587 

373907 

5 

50 

590086 

497 

9(54240 

89 

626445 

586 

373555 

4 

57 

590984 

497 

964187 

89 

626797 

586 

373203 

3 

58 

591282 

497 

904133 

89 

627149 

586 

372851 

2 

59 

591580 

496 

964080 

89 

627501 

585 

372499 

1 

f>0 

591878 

496 

9(54026 

89 

627852 

585 

372148    0 

|      Cosine      | 


Sine        |  |     Cotang. 

67  Degrees. 


1         Tang.       I 


LOGARITHMIC  SINES,  COSINES,  ETC.    (23  Degrees.)      £21 


|    Sine.    |   1).      Cosine      D.      Tan?.      D.   |   Cotang'.   | 

9-5!)  1878 

496 

9-964026 

89 

9-627852    585 

10-372148 

60 

592176 

495 

903972 

89 

628203    585 

371797 

59 

51)2473 

495 

963919 

89 

6285.54 

585 

371446 

58 

592770 

495 

963865 

90 

628905 

584 

371095 

57 

593067 

494 

963811 

90 

629255 

584 

370745 

56 

593363 

494 

963757 

90 

629606 

583 

370394 

55 

593659 

493 

963704 

90 

629956 

583 

370044 

54 

593955 

493 

9636.50 

90 

630306 

583 

369694 

53 

594251 

493 

963596 

90 

630656 

583 

369344 

52 

594547 

492 

963542 

90 

631005 

582 

368995 

51 

594842 

492 

963488 

90 

631355 

582 

368645 

50 

9-595137 

491 

9-963434 

90 

9-631704  ' 

582 

10-368296 

49 

595432 

491 

963379 

90 

632053 

581 

367947 

48 

595727 

491 

963325 

90 

632401 

581 

367599 

47 

596021 

490 

963271 

90 

632750 

581 

367250 

46 

596315 

490 

963217 

90 

633098 

580 

366902 

45 

596609 

489 

963163 

90 

633447 

580 

366553 

44 

596903 

489 

963108 

91 

633795 

580 

366205 

43 

597196 

489 

963054 

91 

634143 

579 

365857 

42 

597490 

488 

962999 

91 

634490 

579 

365510 

41 

597783 

488 

962945 

91 

634838 

579 

365162 

40 

9-598075 

487 

9-962890 

91 

9-635185 

578 

10-364815 

39 

598368 

487 

962836 

91 

635532 

578 

364468 

38 

598660 

487 

962781 

91 

635879 

578 

364121 

37 

598952 

486 

962727 

91 

636226 

577 

363774 

36 

599244 

486 

962672 

91 

636572 

577 

363428 

35 

599536 

485 

962617 

91 

636919 

577 

363081 

34 

599827 

485 

962562 

91 

637265 

577 

362735 

33 

600118 

485 

962508 

91 

637611 

576 

362389 

32 

600409 

484 

962453 

91 

637956 

576 

362044 

31 

600700 

484     962398 

92 

638302 

576 

361698 

30 

9-600990 

484 

9-962343 

92 

9-638647 

575 

10-361353 

29 

601280 

483 

902288 

92 

638992 

575 

361008 

28 

601570 

483 

962233 

92 

639337 

575 

360663 

27 

601860 

482 

962178 

92 

639682 

574 

360318 

26 

602150 

482 

962123 

92 

640027 

574 

359973 

25 

602439 

482 

962067 

92 

640371 

574 

359629 

24 

602728 

481 

962012 

92 

640716 

573 

359284 

23 

603017 

481 

961957 

92 

641060 

573 

35H940 

22 

603305 

481 

961902 

92 

641404 

573 

358596 

21 

603594 

480 

961846 

92 

641747 

572 

358253 

20 

9-603882 

480 

9-961791 

92 

9-642091 

572 

10357909 

19 

604170 

479 

961735 

92 

642434 

572 

357566 

18 

604457 

479 

961680 

92 

642777 

572 

357223 

17 

604745 

479 

961624 

93 

643120 

571 

356880 

16 

605032 

478 

961569 

93 

643463 

571 

356537 

15 

605319 

478 

961513 

93 

643806 

571 

356194 

14 

605606 

478 

961458 

93 

644148 

570 

355852 

13 

605892 

477 

961402 

93 

644490 

570 

355510 

12 

606179 

477 

9G1  34G 

93 

644832 

570 

355168 

11 

606465 

476 

9612i)0 

93 

645174 

569 

354826 

10 

9-606751 

476 

9-961235 

93 

9-645516 

569 

10-354484 

9 

607036 

476 

961179 

93 

645857 

569 

354143 

8 

607322 

475 

961123 

93 

646199 

569 

353801 

7 

607607 

475 

961067 

93 

646540 

568 

353460 

6 

607892 

474 

961011 

93 

646881 

568 

353119 

5 

608177 

474 

960955 

93 

647222 

568 

352778 

4 

608461 

474 

960899 

93 

647562 

567 

352438 

3 

608745 

473 

960843 

94 

647903 

567 

352097 

2 

609029 

473 

960786 

94 

648243 

567 

351757 

6093  13 

473 

900730 

94 

648583 

566 

351417 

0 

Cosine       | 


t 

66  Degrees. 


Cotang. 


Tang. 


222      (24  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.     Sine       D.      Cosine     D.  |   Tang.      D.      CotP.ng.   | 

0 

9-609313 

473 

9-960730 

94 

9-648583 

566 

10-351417 

1 

609597 

472 

9(50674 

94 

648923 

566 

351077 

2 

609880 

472 

960618 

94 

649263 

566 

350737 

3 

610164 

47-2 

960561 

94 

649(502 

5(16 

35031*8 

4 

610447 

471 

960505 

94 

649942 

565 

350058 

5 

6  1  0729 

471 

9(50448 

94 

650281 

505 

349719 

6 

611012 

470 

960392 

94 

650620 

5K5 

349380 

7 

611294 

470 

960335 

94 

650959 

564 

349041 

8 

611576 

470 

960279 

94 

651-297 

5(i4 

348703 

9 

611858 

469 

900222 

94 

65!ti36 

51)4 

3483(54 

10 

612140 

469 

960i65 

94 

651974 

563 

348026 

11 

9-612421 

469 

9-960109 

95 

9-65-2312 

563 

10-347688 

12 

6127i(2 

41)8 

960052 

95 

652650 

5(53 

347350 

13 

61-2983 

4H8 

959995 

95 

652988 

563 

347012 

14 

61*864 

467 

959938 

95 

653326 

562 

.  34(5074 

15 

61:1545 

467 

959882 

95 

653663 

562 

346337 

16 

613825 

4b7 

959825 

95 

654000 

562 

346000 

17 

6I4ID5 

466 

9597(58 

95 

654337 

561 

3456(53 

18 

614385 

466 

959711 

95 

654(574 

561 

345326 

]L) 

614665 

466 

959654 

95 

655011 

561 

344!(89 

20 

614944 

465 

959596 

95 

655348 

561 

344652 

21 

9-615223 

465 

9-959539 

95 

9-655(584 

560 

10-344316 

22 

6i.Wo2 

465 

959482 

95 

65(5020 

560 

343980 

23 

615781 

4ti4 

959425 

95 

656356 

5150 

343644 

24 

616060 

464 

959:568 

95 

656692 

559 

343308 

25 

616338 

464 

959310 

96 

657028 

559 

34-2972 

26 

616616 

463 

959-253 

96 

657364 

559 

342636 

27 

616894 

463 

959195 

96 

657699 

559 

342301 

28 

617172 

4(52 

959138 

96 

658034 

558 

341966 

29 

617450 

462 

959081 

96 

658369 

5.58 

341631 

30 

617727 

462 

959023 

96 

658704 

558 

341-296 

31 

9-618004 

461 

9-958965 

96 

9-659039 

558 

10-340961 

32 

618281 

461 

958908 

9(5 

659373 

557 

340(127 

33 

618558 

461 

958850 

96 

659708 

557 

340-292 

34 

618834 

460 

958792 

96 

660042 

557 

339958 

35 

619110 

460 

958734 

96 

6(50376 

557 

339(524 

36 

619386 

460 

958677 

96 

660710 

556 

339290 

37 

619662 

459 

958619 

96 

661043 

556 

338957 

38 

619938 

459 

958561 

96 

6(51377 

556 

3386-23 

39 

620213 

459 

958503 

97 

661710 

555 

338-290 

40 

620488 

458 

958445 

"97 

662043 

555 

337957 

41 

9-620763 

458 

9-958387 

97 

9-662376 

555 

10-337624 

42 

621038 

457 

958329 

97 

6627(0 

554 

337291 

43 

62*1313 

457 

958271 

97 

663042 

554 

3369.58 

44 

621587 

457 

958213 

97 

663375 

554 

336625 

45 

621861 

456 

958  1.54 

97 

663707 

554 

33(5-293 

46 

622135 

456 

95809(5 

97 

664039 

553 

335961 

47 

622409 

456 

958<C{8 

97 

6(54371 

553 

335(529 

48 

622682 

455 

957979 

97 

664703 

5.r>3 

335297 

49 

622956 

455 

957921 

97 

665035 

553 

334965 

50 

623229 

455 

957863 

97 

665366 

552 

334634 

51 

9-623502 

454 

9-957804 

97 

9-66")6!)7 

552 

10-334303 

52 

623774 

454 

957746 

98 

666!  (29 

552 

333971 

53 

624047 

454 

957687 

98 

666360 

551 

333640 

54 

624319 

453 

957628 

98 

666691 

551 

333309 

55 

624591 

453 

957570 

98 

6671121 

551 

332979 

56 

624863 

453 

957511 

98 

667358 

551 

33-2(548 

57 

625135 

452 

957452 

98 

66768-2 

550 

332318 

98 

625406 

453 

957393 

98 

6(>80I3 

550 

331987 

59 

625677 

45-2 

957335 

98 

668343 

550 

331(557 

60 

625948 

451 

95727(5 

98 

6(58672 

550 

331328 

|   CttUM   I           Siae     |          Cotang.   |        1    Tang-. 

65  Degrees. 

LOGARITHMIC  SINES,  COSINES,  ETC.     (25  Degrees.)      223 


M.     Sine       D.       Cosine     _D.   |   Tang.      D.   |   Coiang.   | 

0 

9-625948 

451 

9-957276 

98 

9-6681  i73 

550 

10-331327 

] 

626219 

451 

957217 

98 

669002 

549 

330998 

o 

62JH90 

451 

957158 

98 

669332 

549 

330668 

3 

6267150 

45J 

957099 

98 

669661 

549 

330339 

4 

627  tO 

450 

957(140 

98 

669991 

548 

331)009 

5 

627:fK) 

450 

9569H1 

98 

670320 

548 

329080 

8 

627  j  70 

449 

956921 

99 

670649 

548 

329351 

7 

6-27*40 

449 

950862 

99 

670977 

548 

329023 

8 

628  j  09 

449 

956803 

99 

671306 

547 

328694 

9 

688378 

448 

956744 

99 

671034 

547 

328366 

1U 

628647 

448 

956684 

99 

67  J  963 

547 

328037 

31 

9  628916 

447 

9-956625 

99 

9-672291 

547 

10-327709 

12 

629  H5 

447 

950566 

99 

672619 

546 

327381 

13 

629453 

447 

956506 

99 

672947 

546 

327053 

14 

629721 

448 

956447 

99 

673274 

546 

326726 

15 

629989 

446 

956387 

99 

673002 

546 

326398 

16 

630257 

446 

951)327 

99 

673929 

545 

326071 

17 

630524 

446 

956263 

99 

674257 

545 

325743 

18 

630792 

445 

956208 

100 

674584 

545 

325416 

19 

63IK59 

445 

956148 

JOO 

674910 

544 

325090 

98 

63  J  326 

445 

956089 

100 

675237 

544 

324763 

21 

9-  S3  1593 

444 

9-956029 

100 

9-675564 

544 

10-324436 

22 

631859 

444 

955969 

100 

675890 

544 

324110 

23 

632125 

444 

955909 

JOO 

676216 

543 

323784 

24 

632392 

443 

955849 

100 

676543 

543 

323457 

25 

632058 

443 

955789 

JOO 

676869 

543 

323131 

26 

632923 

443 

955729 

JOO 

677194 

543 

322806 

27 

633189 

442 

955669 

100 

677520 

542 

322480 

28 

633454 

442 

955609 

100 

677846 

542 

322154 

29 

B337I9 

442 

955548 

100 

678171 

542 

32IH29 

30 

633984 

441 

955488 

100 

678496 

542 

321504 

31  19-634249 

441 

9-955428 

101 

9-678821 

541 

10-321179 

32 

634514 

441) 

9553(58 

101 

679146 

541 

320854 

33 

634778 

440 

955307 

J01 

679471 

541 

3205-29 

34 

635042 

440 

955247 

101 

679795 

541 

320205 

35 

6o5306 

439 

955186 

101 

680120 

540 

319880 

36 

635570 

439 

955126 

101 

680444 

540 

319556 

37 

6.1X34 

439 

955065 

101 

680768 

540 

319232 

38 

636097 

438 

955005 

101 

681092 

540 

318908 

39 

6363150 

438 

954944 

J01 

681416 

539 

318584 

40 

636623 

438 

954883 

101 

681740 

539 

318260 

41  1  9-636886 
42   637148 

437 
437 

9-954823 
954762 

101 
101 

9-682063 
682387 

539 
539 

10-317037 
317613 

43 

637411 

437 

954701 

101 

682710 

538 

317290 

44 

637673 

437 

9.54640 

101 

683033 

538 

316967 

45 

637935 

436 

954579 

101 

683356 

538 

316644 

46 

638197 

436 

954518 

102 

683679 

538 

316321 

47 

638458 

436 

954457 

102 

684001 

537 

315999 

48 

638720 

435 

954396 

102 

684324 

537 

315676 

49 

638981 

435 

954335 

102 

684646 

537 

315354 

50 

639242 

435 

954274 

102 

684968 

537 

315032 

5] 

9-630503 

434 

9-954213 

102 

9-685290 

536 

10-314710 

52 

639764 

434 

954152 

102 

685612 

536 

314388 

53 

641)024 

434 

954090 

102 

685934 

536 

314066 

54 

640284 

433 

954029 

102 

686255 

536 

313745 

55 

640544 

433 

9539<W 

102 

686577 

535 

313423 

56 

640804 

433 

953906 

102 

6H6898 

535 

313102 

57 

641064 

432 

95:1845 

102 

687219 

535 

312781 

58 

641324 

432 

953783 

102 

687540 

535 

312460 

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641584 

432 

953722 

103 

687861 

s:n 

312139 

60 

64J842 

431 

953660 

103 

688182 

534 

311818 

|     Cosine      | 


|       Sine         | 


Cotang. 


64  Degrees. 


I        Tang.       |    M. 


224      (26  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  !    Sine   "<   D.    |   Cosrne     D.   |   Tangr.      D.      Cotan£.   | 

0 

9-041842 

431 

9-953660 

103 

9-088182 

534 

10-311818 

00 

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012101 

431 

953599 

103 

688502 

534 

311498 

99 

s 

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421 

953537 

103 

688823 

534 

311177 

58 

a 

642G18 

430  - 

953475 

103 

689143 

533 

310857 

57 

4 

642877 

430 

953413 

103 

6894(53 

533 

310537 

50 

5 

643135 

430 

953352 

103 

689783 

533 

310217 

55 

6 

643393 

430 

953290 

103 

690103 

533 

309897 

54 

7 

643G59 

429 

953228 

103 

690423 

533 

309577 

53 

8 

643<J08 

429 

95316G 

103 

690742 

532 

309258 

52 

0 

644  1C5 

429 

953104 

103 

6U1CG2 

532 

308938 

51 

10 

644423 

428 

953042 

103 

691381 

532 

308619 

50 

11 

9-644G80 

428 

9-952980 

104 

9-691700 

531 

10-308300 

49 

19 

644936 

428 

952918 

104 

692019 

531 

307981 

48 

13 

645193 

427 

952855 

104 

G92338 

531 

3070(52 

47 

14 

645450 

427 

952793 

104 

692G56 

531 

307344 

40 

15 

64570G 

427 

952731 

104 

692975 

531 

307025 

45 

16 

645962 

42G 

952CC9 

104 

693293 

530 

300707 

44 

17 

646218 

420 

952GC6 

104 

693G12 

530 

306388 

43 

18 

640474 

426 

952544 

104 

693930 

530 

306070 

42 

li) 

646729 

425 

952481 

1C4 

C94248 

530 

305752 

41 

20 

646984 

425 

952419 

104 

G945G6 

529 

305434 

40 

21 

9-647240 

425 

9-952356 

104 

9-694883 

529 

10-305117 

39 

23 

647494 

424 

952294 

104 

695201 

529 

304799 

38 

23 

647749 

424 

952231 

104 

695518 

529 

304482 

37 

34 

648004 

424 

9521G8 

105 

695836 

529 

304164 

30 

25 

648258 

424 

952106 

105 

69G153 

528 

303847 

35 

26 

648512 

423 

952043 

105 

690470 

528 

303530 

34 

27 

G487CG 

423 

951980 

1G5 

69G787 

528 

303213 

33 

'28 

649020 

423 

95J917 

105 

697103 

528 

302897 

32 

P 

649274 

422 

951854 

105 

097420 

527 

302580 

31 

'M 

649527 

422 

951791 

105 

697730 

527 

302264 

30 

31 

9-649781 

422 

9-951728 

105 

9-098053 

527 

10-301947 

29 

32 

650034 

422 

951GG5 

105 

6983G9 

527 

301031 

28 

33 

650287 

421 

951G02 

105 

698G85 

520 

301315 

27 

34 

G50539 

421 

951539 

105 

699001 

52G 

300999 

2rt 

35 

650792 

421 

95J476 

105 

699310 

520 

300684 

25 

36 

651041 

420 

951412 

105 

C99G32 

520 

300308 

24 

37 

051297 

420 

951349 

106 

699947 

520 

300053 

23 

38 

651549 

420 

951286 

106 

7002G3 

525 

299737 

''2 

39 

651800 

419 

951222 

KiO 

700578 

525 

299422 

21 

40 

652052 

419 

951  159 

100 

700893 

525 

299107 

20 

41 

9-052304 

419 

9-951096 

106 

9-701208 

524 

10-298792 

19 

42 

652555 

418 

951032 

100 

701523 

524 

298477 

18 

43 

652806 

418 

950968 

io(f 

701837 

524 

298  1G3 

17 

44 

653057 

418 

950905 

106 

702152 

524 

297848 

10 

45 

653308 

418 

950841 

100 

7024GG 

524 

297534 

15 

40 

653558 

417 

950778 

100 

702780 

523 

297220 

14 

47 

653808 

417 

950714 

100 

71,3095 

523 

29G905 

13 

48 

654059 

417 

950G50 

100 

703409 

523 

29(5591 

12 

49 

654309 

416 

950586 

100 

703723 

523 

290277 

11 

51) 

654558 

416 

950522 

107 

704036 

522 

295904 

10 

51 

G54808 

416 

9-950458 

107 

9-704350 

522 

10-295050 

9 

52 

655058 

416 

950394 

107 

704663 

522 

295337 

8 

53 

655307 

415 

950330 

107 

704977 

522 

295023 

7 

54 

655556 

415 

950200 

107 

7()52!K) 

522 

294710 

0 

55 

655805 

415 

9502(12 

107 

705003 

521 

294397 

5 

56 

656054 

414 

950138 

107 

7059  Hi 

521 

294(184 

4 

57 

050302 

414 

950074 

107 

70G228 

521 

293772 

3 

58 

656551 

414 

950010 

107 

700541 

521 

293459 

2 

59 

050799 

413 

949945 

107 

706854 

521 

293140 

1 

Ml 

657047 

413 

949881 

107 

707100 

520 

292834 

0 

|        Cosine 


|      Cotang. 


Tang. 


63  Degrees. 


LOGARITHMIC  -SIXES,  COSINES,  ETC.    (27  Degrees.)      225 


D.       |      Cosine 


|       D. 


Cotang. 


0 

9-657047 

413 

9-949881 

107 

9-707166 

520 

10-292834 

(50 

1 

657295 

413 

94UW16 

107 

707478 

520 

292522 

59 

2 

657542 

412 

9-W752 

107 

707790 

520 

292210 

58 

3 

657790 

412 

94yti«8 

108 

708102 

520 

291898 

57 

4 

658037 

412 

949023 

108 

708414 

519 

291586 

5fi 

5 

658284 

412 

949558 

108 

708726 

519 

291274 

55 

6 

65t<5:u 

411 

94U494 

108 

709037 

519 

290963 

54 

658778 

411 

94!M29 

108 

709349 

519 

290651 

53 

8 

659025 

411 

94!i:«54 

108 

709660 

519 

290340 

52 

9 

6a«-27  1 

410 

949UOO 

108 

709971 

518 

290029 

51 

III 

6oyoi7 

410 

949235 

108 

710282 

5i8 

289718 

50 

Jl 

9-659763 

410 

9-949170 

108 

9-710593 

518 

10-289407 

49 

1-2 

66U009 

409 

949,i)5 

108 

710904 

518 

289096 

48 

13 

660255 

409 

949040 

108 

711215 

518 

288785 

47 

14 

660501 

4D9 

948975 

108 

711525 

517 

288475 

46 

]5 

6607  40 

409 

948'Jlt) 

108 

711836 

517 

288164 

45 

1-5 

6609'.!  1 

408 

948845 

108 

712146 

517 

287854 

44 

17 

66  J  236 

408 

948780 

109 

712456 

517 

287544 

43 

J^ 

661481 

408 

948715 

109 

712766 

516 

287234 

42 

1!) 

661726 

407 

948650 

109 

713076 

516 

28(5924 

41 

20 

661970 

407 

948584 

109 

713386 

5i6 

286614 

40 

21 

9-662214 

407 

9-948519 

109 

9-713696 

516 

1(1-286304 

39 

22 

662459 

407 

948454 

109 

714005 

516 

285995 

38 

23 

662703 

406 

948388 

109 

714314 

515 

285686 

37 

24 

662946 

406 

948323 

109 

714624 

515 

285376 

36 

2.1 

663190 

406 

948257 

109 

714933 

515 

285067 

35 

2H 

663433 

405 

948192 

109 

715242 

515 

284758 

34 

27 

663677 

405 

948126 

109 

715551 

511 

284449 

33 

28 

663920 

405 

948060 

109 

715860 

514 

284140 

32 

2L> 

6641(53 

405 

947995 

110 

716168 

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283832 

31 

30 

664406 

404 

947929 

110 

716477 

514 

283523 

30 

31 

9-664(548 

404 

9-947863 

110 

9-716785 

514 

10-283215  . 

29 

32 

664891 

404 

947797 

110 

717093 

513 

282907 

28 

33 

665133 

403 

947731 

110 

717401 

513 

282599 

27 

34 

665375 

403 

947665 

110 

717709 

513 

282291 

26 

35 

665617 

403 

947(500 

110 

718017 

513 

281983 

25 

36 

665859 

402 

947533 

110 

718325 

513 

281675 

24 

37 

666100 

402 

947467 

110 

718633 

512 

281367 

23 

38 

666342 

402 

947401 

110 

718940 

512 

281060 

22 

39 

666583 

402 

947335 

110 

719248 

512 

280752 

21 

40 

666824 

401 

947269 

110 

719555 

512 

280445 

20 

41 

9-667065 

401 

9-947203 

110 

9-719862 

512 

10-280138 

19 

42 

667305 

401 

947136 

111 

720169 

511 

279831 

18 

43 

667546 

401 

947070 

111 

720476 

511 

27S524 

17 

44 

667786 

400 

947004 

111 

720783 

511 

279217 

16 

45 

668027 

400 

946937 

111 

721089 

511 

278911 

15 

46 

668267 

400 

946871 

111 

721396 

511 

278604 

14 

47 

668506 

399 

94r>804 

111 

721702 

510 

278298 

13 

48 

668746 

3<>9 

9467.18 

111 

722009 

510 

277991 

12 

49 

668986 

399 

9-)  lit  ,71 

111 

722315 

510 

277685 

11 

50 

669225 

399 

946(504 

111 

722621 

510 

277379 

10 

51 

9-6«9464 

398 

9'946.->38 

iii 

9-722927 

510 

10-277073 

9 

52 

6««:o:i 

398 

946471 

111 

723232 

509 

276768 

8 

53 

6WW42 

398 

946404 

111 

723538 

509 

276462 

7 

54 

-  67(1181 

397 

946337 

111 

723844 

509 

276156 

6 

55 

670419 

397 

94R270 

112 

724149 

509 

275851 

5 

5(5 

670H.T8 

397 

94(5203 

112 

724454 

509 

275546 

4 

57 

67(l89fi 

397 

946136 

112 

724759 

508 

275241 

3 

& 

67li:t4 

396 

94ttO«9 

112 

725065 

508 

274935 

2 

59 

671372 

390 

946002 

112 

725369 

508 

274631 

1 

(HI 

671609 

3% 

945935 

112 

725674 

508 

274326 

0 

Coane 


Cotang. 


I       Tang. 


62  Degrees. 


226      (28  Degrees.)     LOGARITHMIC  SIXES,  COSItfES,  ETC. 


M.  |    Sine       D.  f   Cosine      D.      Tansr.    |   D. 

Cotan?. 

~60" 

0 

9-671609 

396 

9-9459:15 

1J2 

9-7251)74 

508 

10-274326 

1 

671847 

395 

:M58I>8 

112 

7-25979 

508 

2740-21 

59 

2 

67-2084 

3p5 

945800 

112 

7-2(1-284 

507 

273716 

58 

3 

672321 

395 

945733 

112 

726588 

507 

273412 

57 

4 

67-2558 

395 

945666 

112 

726892 

507 

273108 

56 

5 

672795 

394 

945598 

112 

727197 

507 

272803 

.">."> 

6 

673i>32 

394 

945531 

112 

727503 

5U7 

272499 

54 

7 

6732(58 

394 

945464 

113 

727805 

506 

272195 

53 

8 

673505 

394 

945396 

113 

728109 

50(5 

27I8-.JI 

52 

9 

673741 

393 

945328 

1J3 

728412 

5(1(5 

271588 

51 

10 

673977 

393 

945261 

113 

728716 

506 

271284 

50 

11 

9-674213 

393 

9-945193 

113 

9-729020 

506 

10-270980 

49 

B 

674448 

392 

945125 

113 

729323 

505 

270677 

48 

33 

674(584 

392 

945058 

113 

729026 

505 

270374 

47 

34 

f>74!H'J 

392 

'J449UO 

113 

729929 

505 

270071 

46 

35 

67  5  J  55 

392 

944922 

113 

730233 

505 

269767 

45 

16 

675390 

391 

944854 

113 

730535 

505 

269465 

44 

37 

675624 

391 

944786 

113 

730838 

504 

269162 

43 

38 

675859 

391 

944718 

113 

731141 

504 

268859 

42 

39 

676094 

391 

944(550 

113 

731444 

504 

2(58556 

41 

20 

676328 

390 

944582 

314 

73J746 

504 

268254 

40 

21 

9-676562 

390 

9-944514 

314 

9-732048 

504 

30-267952 

39 

22 

676796 

390 

944446 

314 

73-2351 

503 

267649 

38 

23 

677030 

390 

944377 

114 

732653 

503 

267347 

37 

24 

677264 

389 

944309 

314 

732955 

503 

267045 

36 

25 

677498 

389 

944241 

114 

733257 

503 

2(56743 

35 

2<i 

677731 

389 

944172 

314 

733558 

503 

2(56442 

34 

27 

677964 

388 

944104 

314 

733860 

502 

'2(><il40 

33 

28 

67H197 

388 

944036 

314 

734162 

502 

205838 

32 

29 

678430 

388 

943967 

114 

734463 

502 

265537 

31 

30 

678663 

388 

943899 

314 

734764 

502 

265236 

30 

31 

9-678895 

387 

9-943830 

314 

9-735066 

502 

30-264934 

29 

32 

(579128 

387 

943761 

114 

735367 

502 

2f>4(533 

28 

33 

679360 

387 

943693 

115 

735668 

501 

204332 

27 

34 

679592 

387 

943624 

315 

735969 

501 

2(54031 

26 

35 

679824 

386 

943555 

115 

736269 

501 

263731 

25 

36 

680056 

386 

943486 

315 

736570 

501 

263430 

24 

37 

680288 

386 

943417 

135 

736873 

501 

263129 

23 

38 

680519 

385 

943348 

315 

737173 

500 

262829 

22 

39 

680750 

385 

943279 

315 

737473 

500 

2C2529 

21 

40 

680982 

385 

943210 

115 

737773 

500 

262229 

20 

41 

9-681213 

385 

9-943141 

315 

9-738071 

500 

30-261929 

19 

42 

681443 

384 

943072 

315 

438371 

500 

261629 

18 

43 

681674 

384 

943003 

315 

738671 

499 

201329 

17 

44 

681905 

384 

942934 

115 

738971 

499 

261029 

16 

45 

682135 

384 

942864 

315 

739271 

499 

200729 

15 

4f> 

682365 

383 

942795 

316 

739570 

499 

'JH0430 

14 

47 

682595 

383 

942726 

316 

739870 

499 

2(50130 

13 

48 

682825 

383 

942656 

316 

740169 

499 

259831 

12 

49 

683055 

383 

942587 

316 

740468 

498 

259532 

31 

50 

683284 

382 

942517 

116 

740767 

498 

259233 

30 

51 

9-683514 

382 

9-942448 

316 

9-741066 

498 

10-258934 

9 

52 

6R3743 

382 

942378 

116 

741365 

498 

258635 

8 

53 

683972 

382 

942308 

116 

741664 

498 

258336 

7 

54 

684201 

381 

942239 

316 

741962 

4!I7 

258038- 

6 

55 

684430 

381 

942169 

116 

742261 

497 

257739 

5 

56 

684658 

381 

942099 

116 

742559 

497 

257441 

4 

57 

684887 

380 

942029 

116 

742858 

497 

257142 

3 

58 

685115 

380 

941959 

116 

743156 

497 

25f>844 

2 

59 

685343 

380 

941889 

117 

743454 

497 

2511546 

3 

60 

685571 

330 

941819 

117 

743752 

496 

256248 

0 

I      Cosine      | 


I  |     Cotang. 

61  Degrees. 


I         Tang.      |    M. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (20  Degrees.)      227 


M.   | 


Cosine       |        D.      |       Tang.         |       D.        |      Cotang. 


0 

9-tJH.")f)7  1 

380 

9-941819 

117 

9-743752 

496 

10-256248 

1 

685799 

379 

94  J  749 

117 

744050 

496 

255950 

2 

686027 

379 

941(i79 

117 

744348 

496 

255052 

3 

686254 

379 

941609 

117 

744645 

496 

255355 

4 

6864*2 

379 

941539 

117 

744943 

496 

255057 

5 

686709 

378 

9414(19 

117 

745240 

496 

254760 

6 

686936 

378 

941398 

117 

745538 

495 

254462 

7 

687  J  63 

378 

941328 

117 

745835 

495 

254165 

8 

687389 

378 

941258 

117 

746132 

495 

25H868 

9 

687616 

377 

941187 

117 

746429 

495 

253571 

10 

6*7843 

377 

941117 

117 

746726 

495 

253274 

It 

9-688069 

377 

9-941046 

118 

9-747023 

494 

10-252977 

M 

6*8295 

377 

940975 

118 

747319 

494 

252681 

J3 

68*521 

376 

940905 

118 

747616 

494 

252384 

14 

688747 

376 

940*34 

118 

747913 

494 

252087 

15 

688972 

376 

940763 

118 

748209 

494 

251791 

16 

68!)  198 

376 

940(593 

118 

748505 

493 

251495 

n 

689423 

375 

940622 

118 

748801 

493 

251199 

J8 

689648 

375 

940551 

118 

749097 

493 

250903 

19 

689*73 

375 

940480 

118 

749393 

493 

250607 

20 

690098 

375 

940409 

118 

749689 

493 

250311 

21 

9-690323 

374 

9-94(1338 

118 

9-7499*5 

493 

10-250015 

22 

690:>48 

374 

940267 

118 

750281 

492 

249719 

23 

6!)0772 

374 

940196 

118 

750576 

492 

249424 

24 

690996 

374 

940125 

1  19 

750*72 

492 

249128 

23 

6!  1  12-20 

373 

940054 

119 

7511(57 

492 

248833 

2<i 

$11444 

373 

939982 

119 

751462 

492 

24*538 

27 

6911)68 

373 

939911 

119 

751757 

492 

248243 

28 

691892 

373 

939840 

119 

752052 

491 

247948 

29 

692115 

372 

939768 

119 

752347 

491 

247653 

30 

692339 

372 

939G97 

119 

752(542 

491 

247358 

31 

9-692502 

372 

9-939H25 

119 

9-752937 

491 

10-247003 

32 

692785 

371 

939554 

1J9 

753231 

491 

24(5709 

33 

693008 

371 

939482 

119 

753526 

491 

246474 

34 

693231 

371 

939410 

119 

753820 

490 

246180 

35 

693453 

371 

939339 

119 

754115 

490 

245885 

36 

693676 

370 

939267 

120 

754409 

490 

245591 

37 

693898 

370 

939195 

120 

754703 

490 

245297 

38 

694120 

370 

939123 

120 

754997 

490 

245003 

39 

694342 

370 

939052  • 

120 

755291 

490 

244709 

40 

694564 

369 

93P980 

120 

755585 

489 

244415 

41 

9-694786 

369 

9-938908 

120 

9-755878 

489 

10-244122 

42 

695007 

369 

938836 

120 

756172 

489 

243828 

43 

695229 

369 

938763 

120 

756465 

489 

243535 

44 

695450 

368 

938691 

190 

750759 

489 

243241 

45 

69:>f)71 

368 

938619 

120 

757052 

489 

242948 

46 

695892 

368 

938547 

120 

757345 

488 

242(555 

47 

696113 

368 

938475 

120 

757038 

488 

2-12362 

48 

696334 

367 

938402 

121 

757931 

488 

242069 

49 

696554 

387 

93*330 

121 

75*224 

488 

241776 

50 

696775 

367 

938258 

121 

758517 

488 

241483 

51 

9-696995 

367 

9-938185 

121 

9-75*810 

488 

10-241190 

52 

697215 

366 

938113 

121 

759102 

487 

240898 

53 

697435 

366 

•  938040 

121 

759395 

487 

240605 

54 

697654 

366 

937967 

121 

759(587 

4*7 

240313 

55 

697874 

366 

937895 

121 

759979 

487 

240021 

56 

698094 

365 

937822 

121 

760272 

4?7 

239728 

57 

698313 

365 

937749 

121 

7605(54 

487 

239436 

58 

698532 

365 

937676 

121 

760856 

.486 

239]  44 

59 

698751 

365 

93"604 

121 

761148 

486 

238852 

60 

698970 

364 

937531 

121 

761439 

486 

23856] 

I      Cosine      | 


Sme         |  |       Cotang.     | 

60  Degrees. 


I       Tang.      |    M. 


228      (30  Degrees.)     LOGARITHMIC  SIXES,  COSINES,  ETC. 


M. 

Sme       D.     Cosine       D.      Tang. 

D.   |   Cotanjy.   | 

0 

9-698970 

364 

9-907531 

121 

9-761439 

486 

10--2385I51 

60 

1 

699189 

364 

9374.58 

J22 

761731 

486 

238269 

59 

2 

699407 

364 

937385 

122 

762023 

486 

237977 

58 

3 

699626 

364 

937312 

122 

702314 

486 

2376*6 

57 

4 

699844 

363 

937238 

122 

762(106 

485 

237394 

56 

5 

700062 

363 

937165 

199 

762897 

485 

237103 

55 

6 

700280 

363 

937092 

122 

7031  88 

485 

236812 

54 

7 

700498 

363 

937019 

122 

763479 

485 

2365-21 

53 

8 

700716 

363 

93(5946 

122 

763770 

485 

23(5230 

;>2 

9 

700933 

362 

93(5872 

122 

7(54061 

485 

235939 

51 

10 

701151 

362 

93(5799 

122 

764352 

484 

235648 

50 

11 

9-701368 

362 

9-936725 

122 

9-764(543 

484 

10-235357 

49 

12 

701585 

362 

936652 

123 

7(54933 

484 

235067 

48 

13 

701802 

361 

936578 

123 

765824 

484 

234776 

47 

14 

702019 

361 

936505 

123 

765514 

484 

23448(5 

46 

15 

702236 

361 

936431 

123 

7(55805 

484 

234195 

45 

16 

702452 

361 

936357 

123 

76C.095 

484 

233905 

44 

17 

702669 

360 

936284 

123 

7(5(5385 

483 

233615 

43 

18 

702885 

360 

936210 

1*3 

7(5(5(575 

483 

233325 

42 

19 

703101 

360 

936136 

123 

766U65 

483 

233035 

41 

20 

703317 

360 

936062 

123 

767255 

483 

23-2745 

40 

21 

9-703533 

359 

9-935988 

123 

9-767545 

483 

10-232455 

39 

22 

703749 

359 

935914 

123 

767834 

483 

'23-2166 

38 

23 

703964 

359 

935840 

123 

768124 

482 

231876 

37 

24 

7(i41V9 

359 

935766 

124 

768413 

482 

231587 

36 

25 

704395 

359 

935692 

124 

768703 

482 

231297 

35 

20 

704610 

358 

935618 

124 

768992 

482 

231008 

34 

27 

704825 

358 

935543 

124 

769281 

482 

230719 

33 

28 

705040 

358 

935469 

124 

769570 

482 

230430 

32 

29 

705254 

358 

935395 

124 

7698GO 

481 

230140 

31 

30 

705469 

357 

935320 

124 

770148 

481 

229852 

30 

31 

9-705683 

357 

9-935246 

124 

9-770437 

481 

10-229563 

29 

32 

705898 

357 

935171 

124 

770726 

481 

229274 

28 

33 

706112 

357 

935097 

124 

771015 

481 

228985 

27 

34 

706326 

356 

935022 

124 

771303 

481 

228697 

26 

35 

706539 

356 

934948 

124 

771592 

481 

228408 

25 

30 

706753 

356 

934873 

124 

771880 

480 

228120 

24 

37 

706967 

356 

934798 

125 

772168 

480 

227832 

23 

38 

707180 

355 

934723 

125 

772457 

480 

227543 

22 

39 

707393 

355 

934  (54  9 

125 

772745 

480 

227255 

21 

40 

707606 

355 

934574 

125 

773033 

480 

226967 

20 

41 

9-707819 

355 

9-934499 

125 

9-773321 

480 

10-226679 

19 

42 

708032 

354 

934424 

125 

773608 

479 

22(5392 

18 

43 

708245 

354 

934349 

125 

7738!  )f> 

479 

22(5104 

17 

44 

708458 

354 

934274 

125 

774184 

479 

2-25816 

16 

45 

708670 

354 

934199 

125 

774471 

479 

225529 

15 

46 

7(18882 

353 

934123 

125 

774759 

479 

225241 

14 

47 

709094 

353 

934048 

125 

77504(5 

479 

2-24954 

13 

48 

70930fl 

353 

933973 

125 

775333 

479 

224(567 

12 

49 

709518 

353 

933898 

126 

775621 

478 

2-24379 

11 

50 

709730 

353 

933822 

126 

775908 

478 

224092 

10 

51 

9-709941 

352 

9-9:0747 

.  126 

9-776195 

478 

10-223805 

9 

52 

710153 

352 

933671 

126 

77(5482 

478 

223518 

8 

53 

710364 

352 

933596 

126 

776769 

478 

223231 

7 

54 

710575 

352 

933520 

126 

777055 

478 

222945 

6 

55 

710786 

351 

933445 

126 

777342 

478 

222658 

5 

56 

71  01197 

351 

933369 

126 

7776-28 

477 

222372 

4 

57 

711208 

351 

933293 

126 

777915 

477 

222085 

3 

58 

711419 

351 

933217 

126 

778201 

477 

221799 

o 

511 

711629 

350 

9:0141 

196 

778487 

477 

221512 

1 

GO 

71  1839 

350 

9330(56 

126 

778774 

477 

2-2  J  -2-26 

0 

I        Sine 


|  •    |      Cotanf*.     I 

59  Degrees. 


J    M. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (31  Degrees.)      229 


M.    I         Sine 


D.       |       Cosine       |      D.       |       Trng.        |       D.       |      Colang.       I 


o 

9-711839 

350 

9-933066 

126 

9-778774 

477 

10-221226 

60 

712050 

350 

932990 

127 

7790(50 

477 

220940 

59 

2 

712260 

350 

932914 

127 

779346 

476 

220654 

58 

3 

712469 

349 

932838 

127 

779632 

476 

220368 

57 

4 

712679 

349 

932762 

127 

779918 

476 

220082 

56 

5 

712889 

349 

932685 

127 

7802<»3 

476 

219797 

55 

6 

713098 

349 

932609 

127 

780489 

476 

219511 

54 

7 

713303 

349 

932533 

127 

780775 

476 

2111225 

53 

8 

713517 

348 

932457 

127 

781060 

476 

218940 

52 

9 

71  3726 

348 

932380 

127 

781346 

475 

218654 

51 

10 

713935 

348 

932304 

127 

781631 

475 

2.18369 

50 

11 

9-714144 

348 

9-932228 

127 

9-781916 

475 

10-218084 

49 

12 

714352 

347 

932151 

127 

782201 

475 

217799 

48 

13 

714561 

347 

932075 

128 

782486  ' 

475 

217514 

47 

14 

714769 

347 

931998 

128 

782771 

475 

217229 

46 

15 

714978 

347 

931921 

128 

783056 

475 

2-16944 

45 

Ifi 

715186 

347 

931845 

128 

783341 

475 

216659 

44 

17 

715394 

346 

9317(58 

128 

783626 

474 

2J6374 

43 

18 

715(502 

346 

931691 

128 

783910 

474 

216090 

42 

19 

715809 

346 

931(514 

128 

784195 

474 

215805 

41 

20 

716017 

346 

931537 

128 

784479 

474 

215523 

40 

21 

9-716224 

345 

9-931460 

128 

9-7847«4 

474 

10-215236 

39 

22 

716432 

345 

931383 

128 

785048 

474 

214952 

38 

2:} 

716639 

345 

931308 

128 

785332 

473 

214668 

37 

24 

716846 

345 

931229 

129 

785616 

473 

214384 

36 

25 

717053 

345 

931152 

129 

785900 

473 

214100 

35 

2(5 

717259 

344 

931075 

129 

786184 

473 

213816 

34 

27 

717466 

344 

930998 

129 

786468 

473 

213532 

33 

28 

717673 

344 

930921 

J29 

78(5752 

473 

213248 

32 

29 

717879 

344 

930843 

129 

787036 

473 

2129(54 

31 

30 

718085 

343 

930766 

129 

787319 

472 

2J2<581 

30 

31 

9-718291 

343 

9-930r>88 

129 

9-787603 

472 

10-212397 

29 

32 

718497 

343 

93061  1 

129 

787886 

472 

212114 

28 

33 

718703 

343 

93:1533 

129 

788170 

472 

211830 

27 

34 

718909 

343 

930456 

129 

788453 

472 

211547 

26 

35 

719114 

342 

930378 

129 

788736 

472- 

211264 

25 

30 

719320 

342 

930300 

130 

789019 

472 

210981 

24 

37 

719525 

342 

930223 

130 

789302 

471 

210698 

23 

38 

719730 

342 

930145 

130 

789585 

471 

210415 

22 

39 

719935 

341 

930067 

130 

789868 

471 

210132 

21 

40 

720140 

341 

929989 

130 

•790151 

471 

209849 

20 

41 

9-720345 

341 

9-929911 

130 

9-790433 

471 

10-209567 

19 

42 

720549 

341 

929833 

130 

790716 

471 

209234 

18 

43 

720754 

340 

929755 

130 

790999 

471 

209001 

17 

44 

720958 

340 

929677 

130 

791281 

471 

208719 

J6 

45 

721162 

340 

929599 

130 

791563 

470 

208437 

15 

46 

721366 

340 

929521 

130 

791846 

470 

208154 

14 

47 

721570 

340 

9211442 

130 

792128 

470 

207872 

13 

48 

721774 

339 

929364 

131 

792410 

470 

207590 

12 

49 

721978 

339 

"  929286 

131- 

792692 

470 

207308 

11 

51) 

722181 

339 

929207 

131 

792974 

470 

20702G 

10 

51 

•j-722385 

339 

9-929129 

131 

9-793256 

470 

10-206744 

9 

52 

722588 

339 

929050 

131 

793538 

4(59 

206462 

8 

53 

722791 

338 

928972 

131 

793819 

469 

206181 

7 

54 

722994 

338 

928893 

131 

794101 

469 

205899 

6 

55 

723197 

338 

928815 

131 

7941183 

469 

205G17 

5 

5(5 

723400 

338 

928736 

131 

7(146(54 

469 

205336 

4 

57 

"723603 

337 

928657 

131 

794945 

469 

205055 

3 

58 

723805 

337 

928578 

131 

795227 

469 

204773 

2 

59 

724007 

337 

928499 

131 

795508 

468 

204492 

1 

60 

724210 

337 

928420 

131 

795789 

468 

204211  '  0 

|       Cosine       | 


Sine         |  |      Cotang.     | 

53  Decrees, 


Tang.        )   M. 


230      (32  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.   |       Sine 


Cosine      |       D.       |      Tang.       |        D.       |       Cota 


0 

9-724210 

337 

9-92H420 

132 

9-795789 

468 

10-204211 

1 

724412 

337 

928342 

132 

796070 

468 

203930 

2 

724614 

336 

928263 

132 

790351 

468 

203649 

3 

724816 

336 

928183 

132 

796632 

468 

203368 

4 

725017 

336 

928104 

132 

796913 

468 

203087 

5 

725219 

336 

928025 

132 

797194 

468 

202806 

6 

725420 

335 

927946 

132 

797475 

468 

202525 

7 

725622 

335 

927867 

132 

797755 

468 

202245 

8 

725823 

335 

927787 

132 

798038 

467 

201964 

9 

726024 

335 

927708 

132 

798316 

467 

201684 

10 

720225 

335 

927629 

132 

798596 

467 

201404 

11 

9-726426 

334 

9-927549 

132 

9-798877 

467 

10-201123 

12 

726626 

334 

927470 

133 

799157 

467 

200843 

13 

726827 

334 

927390 

133 

799437 

467 

200563 

14 

727027 

334 

927310 

133 

799717 

467 

200283 

15 

727228 

334 

927231 

133 

799997 

466 

200003 

16 

727428 

333 

92715% 

133 

800277 

466 

199723 

17 

727628 

333 

927071 

133 

800557 

466 

199443 

18 

727828 

333 

926991 

133 

800836 

466 

199164 

19 

728027 

333 

926911 

133 

801116 

466 

198884 

20 

728227 

333 

920831 

133 

801396 

4(56 

198604 

21 

9-728427 

332 

9-926751 

133 

9-801675 

466 

10-198325 

22 

728626 

332 

926671 

133 

8C1955 

468 

198045 

23 

728825 

332 

926591 

133 

802234 

465 

]  97766 

24 

729IJ24 

332 

926511 

134 

802513 

465 

197487 

25 

729223 

331 

926431 

134 

802792 

465 

197208 

26 

729422 

331 

926351 

134 

803072 

465 

196928 

27 

729621 

331 

926270 

134 

803351 

465 

196649 

28 

729820 

331 

926190 

134 

803030 

465 

196370 

29 

730018 

330 

926110 

134 

803908 

465 

196092 

30 

730216 

330 

926029 

134 

804187 

465 

195813 

31 

9-730415 

330 

9-925949 

134 

9-804466 

464 

10-195534 

32 

730613 

330 

925868 

134 

804745 

464 

195255 

33 

730811 

330 

925788 

134 

805023 

464 

194977 

34 

731009 

329 

925707 

134 

805302 

464 

194698 

35 

731206 

329 

925626 

134 

805580 

464 

194420 

3(5 

731404 

329 

925545 

135 

805859 

464 

194141 

37 

731(502 

329 

925465 

135 

806137 

464 

193863 

38 

731799 

329 

925384 

135 

806415 

463 

193585 

39 

73  J  996 

328 

925303 

135 

806693 

463 

193307 

40 

732193 

328 

925222' 

135 

806971 

463 

193029 

41 

9-732390 

328 

9-925141 

135 

9-807249 

463 

10-192751 

42 

732587 

328 

925060 

135 

807527 

463 

192473 

43 

732784 

328 

924979 

135 

807805 

463 

190195 

44 

732980 

327 

924897 

135 

808083 

463 

101917 

45 

733177 

327 

924P16 

135 

808361 

463 

191639 

46 

733373 

327 

9247H5 

136 

808638 

4C2 

1913G2 

47 

733569 

327 

924654 

136 

808916 

462 

191084 

48 

733765 

327 

924572 

136 

809193  . 

4C2 

190807 

49 

733961 

326 

924491 

136 

809471 

462 

190529 

50 

734157 

326 

924409 

136 

809748 

462 

190252 

51 

9-734353 

326 

9-924328 

136 

9-810025 

462 

10-189975 

52 

73-1549 

326 

92-1246 

136 

810302 

4C2 

189698 

53 

734744 

325 

924164 

136 

810580 

462 

189420 

54 

734939 

325 

924083 

136 

810857 

462 

1P9143 

55 

735135 

325 

924001 

136 

811134 

461 

18P866 

56 

735330 

325 

5)23919 

136 

811410 

461 

18P590 

57 

735525 

325 

923837 

136 

811687 

461 

18P313 

58 

7?5719 

324 

9237.-).-) 

137 

811964 

461 

188036 

59 

735914 

324 

923673 

137 

812241 

461 

187759 

<*) 

736109 

324 

923591 

137 

812517 

461 

187483 

|     Cotan-. 


I         Tta%.      \    M. 


67  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (33  Degrees.)      231 


M.  '    Sine    }   D.     Cosine   |   D.      Tang.      D.   |   Cotang.   1 

0 

9-730109 

324 

9-923591 

137 

9-812517 

461 

10-187482 

00 

1 

730303 

324 

923509 

137 

812794 

401 

187206 

59 

2 

736498 

324 

923427 

137 

813070 

461 

180930 

58 

3 

730692 

323 

923345 

137 

813347 

460 

186053 

57 

4 

736886 

323 

923263 

J37 

813623 

460 

180377 

56 

5 

737080 

323 

923181 

137 

813899 

460 

180101 

55 

6 

737274 

323 

923098 

137 

814175 

460 

185825 

54 

7 

737467 

323 

923016 

137 

814452 

460 

185548 

53 

8 

737(501 

322 

922933 

137 

8147-28 

460 

185272 

52 

9 

737855 

322 

922851 

137 

815004 

460 

184996 

51 

10 

738048 

322 

922768 

138 

815279 

460 

184721 

50 

]] 

9-738241 

322 

9-922086 

138 

9-815555 

459 

10184445 

49 

12 

738434 

322 

922603 

138 

815831 

459 

1H4109 

48 

13 

738627 

321 

922520 

138 

816107 

459 

183893 

47 

14 

738820 

321 

922438 

138 

816382 

459 

183618 

46 

15 

739013 

321 

922355 

138 

816658 

459 

183342 

45 

16 

739206 

321 

922272 

138 

816933 

459 

183007 

44 

17 

739398 

321 

922189 

138 

817209 

459 

182791 

43 

18 

739590 

320 

922  1G6 

138 

817484 

459 

182516 

42 

19 

739783 

320 

9221,23 

138 

817759 

459 

182241 

41 

20 

739975 

320 

921940 

138 

818035 

458 

181965 

40 

21 

9-740167 

320 

9-921857 

139 

9-818310 

458 

10-181690 

39 

22 

740359 

320 

921774 

139 

818585 

458 

181415 

38 

23 

740550 

319 

921691 

139 

818800 

458 

181140 

37 

24 

740742 

319 

92  J  607 

139 

819135 

458 

180865 

36 

25 

740934 

319 

921524 

139 

819410 

458 

180590 

35 

20 

741125 

319 

921441 

139 

819684 

458 

180316 

34 

27 

741316 

319 

921357 

139 

819959 

458 

180041 

33 

28 

74  J  508 

318 

921274 

139 

820234 

458 

179766 

32 

29 

741699 

318 

921  190 

139 

820508 

457 

179492 

31 

30 

741  889 

318 

921107 

139 

820783 

457 

179217 

30 

31 

9-742080 

318 

9-921023 

139 

9-821057 

457 

10-178943 

29 

32 

742271 

318 

920939 

140 

821332 

457 

178668 

28 

33 

7424G2 

317 

92C856 

140 

8210(,6 

457 

178394 

27 

34 

742052 

317 

920772 

140 

821880 

457 

178120 

26 

35 

742842 

317 

920G88 

140 

822154 

457 

177846 

25 

30 

743033 

317 

92G6G4 

140 

822429 

457 

177571 

24 

37 

74:>223 

317 

92G520 

140 

822703 

457 

177297 

23 

38 

743413 

316 

920436 

140 

822977 

456 

177023 

22 

39 

743602 

316 

920352 

140 

823250 

456 

176750 

21 

40 

743792 

316 

D202C8 

140 

823524 

456 

176476 

20 

41 

9-743982 

316 

9-920184 

140 

9-823798 

456 

10-176202 

19 

42 

744171 

316 

920099 

140 

824072 

456 

175928 

18 

43 

744361 

315 

921015 

140 

824245 

456 

175055 

17 

44 

744550 

315 

919931 

141 

824619 

456 

17o381 

16 

45 

744739 

315 

919846 

141 

824feD3 

456 

175107 

15 

40 

744928 

315 

919702 

141 

825JC6 

456 

174834 

14 

47 

745117 

315 

9J9077 

141 

825439 

455 

174501 

13 

48 

745306 

314 

919593 

141 

825713 

455 

174287 

12 

49 

745494 

314 

919508 

141 

825986 

455 

174014 

11 

50 

745683 

314 

919424 

141 

826259 

455 

173741 

10 

51 

9-745871 

3J4 

9-919339 

141 

9-826532 

455 

10-173468 

9 

52 

746059 

314 

919254 

141 

820805 

455 

173195 

8 

53 

746248 

313 

919109 

141 

827078 

455 

172922 

7 

54 

740436 

313 

919085 

141 

827351 

455 

172649 

6 

55 

746624 

313 

919000 

141 

827624 

455 

172376 

5 

56 

746812 

313 

9J8915 

142 

827897 

454 

172103 

4 

57 

746999 

313 

918830 

142 

828170 

454 

171!:30 

3 

X 

747187 

312 

918745 

142 

828442 

454 

171558 

2 

59 

747374 

312 

918659 

142 

828715 

454 

171285 

j 

60 

747562 

312 

918574 

142 

828987 

454 

171013  l  0 

I      Coeine        I 


Sine        | 


Cotang. 


Tnng. 


56  Degrees 


232       (34  Degrees.)    LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Sine       D.     Cosine      D.   |  Targ.    f   D. 

Golan*. 

0 

9-7475G2 

312 

9-918574 

142 

9-828987 

454 

10-171013 

1 

747749 

312 

i)  18489 

142 

829200 

454 

170740 

2 

747J36 

312 

918404 

142 

829532 

454 

170408 

3 

71dl23 

311 

918318 

J42 

82-.i*<  15 

454 

170195 

4 

746310 

311 

9  J  8233 

142 

Kin  177 

454 

169923 

5 

748497 

311 

918147 

142 

83u3-19 

453 

169651 

0 

746683 

311 

91riM>2 

142 

83(>021 

453 

10!  (379 

7 

748870 

311 

917976 

143 

83urt93 

453 

109107 

8 

74JU56 

310 

917891 

143 

8311t>5 

453 

168835 

9 

74J243 

310 

9178u5 

143 

831437 

453 

16KVJ3 

10 

7494-29 

3JO 

917719 

143 

831709 

453 

168291 

11 

9-749615 

310 

9-917634 

143 

9-831981 

453 

10-168019 

12 

74J3J1 

310 

917548 

143 

832253 

453 

107747 

13 

749987 

309 

917462 

143 

832525 

453 

107475 

14 

750172 

309 

917376 

143 

832796 

453 

107204 

15 

750358 

309 

9J7290 

143 

833068 

4f>2 

100932 

10 

750543 

309 

9  J  7204 

143 

833339 

452 

-  KiOOOl 

17 

750729 

309 

917118 

144 

833011 

452 

100389 

18 

750914 

308 

917032 

144 

833882 

452 

100118 

19 

751J99 

308 

916946 

144 

834154 

452 

105846 

20 

751284 

308 

916859 

144 

834425 

452 

105575 

21 

9-751469 

308 

9-910773 

144 

9-834096 

452 

10-105304 

2-2 

751G54 

308 

910087 

144 

834907 

452 

1  65033 

23 

751839 

308 

910000 

144 

835238 

452 

104702 

24 

752u23 

307 

916514 

144 

835509 

452 

104491 

25 

752-208 

307 

916427 

144 

835780 

451 

104220 

26 

752392 

307 

916341 

144 

830051 

451 

163949 

27 

752576 

307 

91i,254 

144 

830322 

451 

103078 

28 

752760 

307 

916167 

145 

830593 

451 

163407 

29 

752944 

306 

916081 

145 

830804 

451 

163136 

30 

753128 

306 

915994 

145 

837134 

451 

162866 

31 

9-753312 

306 

9-915907' 

145 

9-837405 

451 

10-162595 

32 

753495 

306 

915820 

145 

837675 

451 

162325 

33 

753679 

306 

915733 

145 

837946 

451 

102054 

34 

753862 

305 

915646 

145 

838216 

451 

101784 

35 

754046 

305 

915559 

145 

838487 

450 

101513 

36 

754229 

305 

915472 

145 

838757 

450 

101243 

37 

754412 

305 

915385 

145 

839027 

450 

100973 

38 

754595 

305 

915297 

145 

839297 

450 

100703 

31) 

754778 

304 

915210 

145 

839508 

450 

100432 

40 

754960 

304 

915123 

146 

839838 

450 

160162 

41 

9-755143 

304 

9-915035 

146 

9-840108 

450 

10-159892 

4-2 

755326 

304 

914948 

146 

840378 

450 

159622 

43 

755508 

304 

914860 

146 

840047 

450 

159353 

44 

755690 

304 

914773 

146 

840917 

449 

159083 

45 

755872 

303 

914685 

146 

841187 

449 

158813 

46 

756054 

303 

914598 

146 

841457 

449 

158543 

47 

756236 

303 

914510 

146 

841726 

449 

158274 

48 

756418 

303 

914422 

146 

841996 

449 

158004 

49 

756600 

303 

914334 

146 

842206 

449 

157734 

90 

756782 

302 

914246 

147 

842535 

449 

157465 

51 

9-756963 

302 

9-914158 

147 

9-842805 

449 

10157195 

a 

757144 

302 

914070 

147 

843074 

449 

150926 

53 

757326 

302 

913982 

147 

843343 

449 

150657 

54 

757307 

302 

913894 

147 

843612 

449 

156388 

55 

757688 

301 

913806 

147 

843882 

448 

156118" 

56 

757869 

301 

913718 

147 

844151 

448 

155849 

57 

758050 

301 

913630 

147 

844420 

448 

155580 

58 

758230 

301 

913541 

147 

844689 

448 

155311 

r>u 

758411 

3*)1 

91:5453 

147 

844958 

448 

15.-JU42 

90 

758591 

301 

913365 

147 

845227 

448 

154773 

!       Cosine       | 


|       S.ne 


Cotang. 


I         Tang. 


M. 


55  Degrees.  , 


LOGARITHMIC  SINES,  COSINES,  ETC.    (35  Degrees.)      233 


Sine        (       D.       |      Cosine        1       D.       |       Tang-.         |       D.       |     Golan?. | 


0 

9-758591 

301 

9-913365 

147 

9.845227 

448 

10-154773 

(k) 

1 

758772 

300 

913276 

147 

845496 

448 

154504 

59 

2 

758952 

300 

913187 

148 

845764 

448 

154236 

58 

3 

759132 

300 

913099 

148 

846033 

448 

153967 

57 

4 

75U312 

300 

913010 

148 

846302 

448 

153698 

58 

5 

75U492 

300 

912922 

148 

846570 

447 

153430 

5.) 

.  6 

7.19G72 

299 

912d33 

148 

846839 

447 

153161 

54 

7 

75985-2 

299 

912744 

148 

847107 

447 

152893 

53 

8 

760031 

299 

912655 

148 

847376 

447 

152624 

52 

9 

700211 

299 

912566 

148 

847644 

447 

152356 

51 

10 

760390 

299 

912477 

148 

847913 

447 

152087 

50 

11 

9-7G0569 

298 

9-912388 

148 

9.848181 

447 

10-151819 

49 

12 

760748 

298 

D12299 

149 

848449 

447 

151551 

48 

13 

760927 

298 

912210 

149 

848717 

447 

151283 

47 

14 

761106 

298 

912121 

W9 

848986 

447 

151014 

46 

15 

761-285 

298 

9121)31 

149 

84<J254 

447 

150746 

45 

16 

761464 

2J8 

9111)42 

149 

849522 

447 

150478 

44 

17 

761642 

297 

911S53 

149 

849790 

446 

15u210 

43 

18 

761821 

297 

91  1763 

149 

850(158 

446 

149942 

42 

19 

761999 

297 

911674 

149 

850325 

446 

149G75 

41 

20 

762177 

297 

911584 

149 

850593 

446 

149407 

40 

21 

9-762356 

297 

9-911495 

149 

9850r'61 

446 

10-149139 

39 

22 

762534 

296 

911405 

149 

851129 

446 

148871 

38 

23 

762712 

296 

911315 

150 

851396 

446 

148G04 

37 

24 

762889 

296 

911226 

150 

851664 

446 

148336 

3(> 

25 

763067 

296 

9imo 

150 

851931 

446 

148069 

35 

26 

763245 

296 

911040 

150 

852199 

446 

147801 

34 

27 

763422 

296 

910956 

150 

852466 

446 

147534 

33 

28 

763600 

295 

910866 

150 

852733 

445 

147267 

32 

29 

763777 

295 

91  0776 

150 

853001 

445 

1461)99 

31 

30 

763954 

295 

910686 

150 

853268 

445 

146732 

30 

31 

9-764131 

295 

9-910596 

150 

9-853535 

445 

10-146465 

29 

32 

764308 

295 

910506 

150 

853802 

445 

146198 

28 

33 

704485 

294 

910415 

150 

854069 

445 

145:>31 

27 

34 

764662 

294 

910325 

51 

8543J6 

445 

145664 

26 

35 

764838 

294 

910235 

51 

854603 

445 

145397 

25 

36 

765015 

294 

910144 

51 

854870 

445 

145130 

24 

37 

765191 

294 

910054 

51 

855137 

445 

144863 

23 

38 

765367 

294 

909963 

51 

855404 

445 

144596 

22 

39 

765544 

293 

909873 

51 

855671 

444 

144329 

21 

40 

765720 

293 

909782 

51 

'  855938 

444 

144062 

20 

41 

9-765896 

293 

9-909691 

51 

9-856204 

444 

10-143796 

19 

42 

766072 

293 

909601 

51 

856471 

•444 

143529 

18 

43 

766247 

293 

909510 

51 

856737 

444 

143263 

17 

44 

766423 

293 

909419 

151 

8570;t4 

444 

142996 

6 

45 

766598 

292 

909328 

152 

857270 

444 

142730 

5 

46 

766774 

292 

90U237 

152 

857537 

444 

142463 

4 

47 

766949 

292 

909146 

152 

857803 

444 

142197 

13 

48 

767124 

OIJO 

909055 

152 

858069 

444 

141931 

12 

49 

767300 

292 

908964 

152 

858336 

444 

141664 

H 

50 

767475 

291 

908873 

152 

858602 

443 

141398 

10 

51 

9-767649 

291 

9-908781 

152 

9-858863 

443 

10-141  1.72 

9 

52 

767824 

291 

90H690 

152 

85U134 

443 

140866 

8 

53 

767999 

291 

908599 

152 

859400 

443 

140600 

7 

54 

768173 

291 

908507 

152 

859666 

443 

140334 

6 

55 

768348 

290 

908416 

153 

859932 

443 

140068 

5 

56 

768522 

290 

908324 

153 

860198 

443 

139802 

4 

57 

768097 

2!»0 

908233 

153 

860464 

443 

13<).r>36 

3 

58 

768871 

2!K) 

908141 

153 

860730 

443 

139270 

2 

59 

769045 

2!H) 

908049 

153 

860995 

443 

139005 

1 

00 

769219 

290 

907958 

153 

861261 

443 

138739 

0 

Sine       | 


I      Cotang.     | 


I        Tang. 


54  Degrees. 


234      (36  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.  |    Sine    |   D. 

Cosine   |   D.   |   Tan?.   |   D.     Cotang.  j 

0 

9-769219- 

290 

9-907958 

153 

9-861261 

443 

10-138739 

60 

1 

709393 

289 

907806 

153 

861527 

443 

138473 

59 

2 

709566 

289 

907774 

153 

861792 

442 

138208 

58 

3 

7G;)740 

289 

907082 

253 

862058 

442 

137942 

57 

4 

709913 

289 

907590 

153 

862323 

442 

137077 

50 

5 

770087 

289 

907498 

153 

802589 

44t> 

137411 

55 

6 

771)200 

288 

907-406 

153 

882854 

442 

137146 

54 

7 

770433 

288 

907314 

154 

803119 

442 

136881 

53 

8 

770006 

288 

907222 

154 

8G3385 

442 

136015 

59 

9 

'  770779 

288 

907129 

154 

803050 

442 

136350 

51 

10 

770952 

288 

907037 

154 

803915 

442 

130085 

50 

11 

9-771125 

288 

9-906945 

154 

9-804180 

442 

10-135820 

49 

12 

77121)8 

287 

906852 

154 

804445 

442 

135555 

48 

13 

771470 

287 

900700 

154 

804710 

442 

135290 

47 

14 

77  J  043 

287 

906607 

154- 

864975 

441 

135025 

40 

15 

771815 

287 

900575 

154 

805240 

441 

134700 

45 

16 

771987 

287 

906482 

154 

805.505 

441 

134495 

44 

17 

772159 

287 

900389 

155 

805770 

441 

134230 

43 

18 

772331 

280 

906290 

155 

800035 

441 

J33'..G5 

42 

19 

772503 

2F6 

90(W04 

155 

860300 

441 

133100 

41 

20 

772675 

286 

900J11 

155 

800564 

441 

133436 

40 

21 

9'772847 

28G 

9-900018 

155 

9-866829 

441 

10-133171 

39 

22 

773018 

286 

905925 

155 

867094 

441 

132906 

38 

23 

773190 

286 

905832 

155 

367358 

441 

132642 

37 

24 

773301 

285 

905739 

155 

867623 

441 

132377 

30 

25 

773533 

285 

905645 

155 

8o7S87 

441 

132113 

35 

20 

773704 

285 

905552 

155 

8(58152 

440 

131848 

34 

27 

773875 

285 

905459 

155 

808416 

44U 

131584 

33 

28 

77J04G 

285 

905366 

156 

80«uoO 

440 

131320 

32 

99 

774217 

285 

905272 

156 

808945 

440 

131055 

:n 

30 

774388 

284 

905179 

156 

869209 

440 

13(1791 

30 

31 

9-774558 

284 

9-905085 

156 

9-869473 

440 

10-130527 

2<) 

32 

774729 

284 

904992 

156 

809737 

440 

J  30203 

28 

33 

774899 

284 

904898 

156 

870001 

440 

12!l!l;»9 

87 

34 

775070 

284 

904804 

156 

870205 

440 

129735 

28 

35 

775240 

284 

904711 

156 

870529 

440 

129471 

25 

31. 

77:>410 

283 

904617 

156 

870793 

440 

129207 

24 

37 

775580 

283 

904523 

156 

871057 

440 

128943 

23 

38 

775750 

283 

904429 

157 

871321 

440 

128679 

22 

39 

7751120 

283 

904335 

157 

871585 

440 

128415 

21 

40 

776090 

283 

90424] 

157 

871849 

439 

128151 

20 

41 

9-776259 

283 

9-904147 

157 

9-872112 

439 

10-127888 

19 

42 

77G429 

282  ' 

904053 

157 

872376 

439 

127624 

J8 

43 

770598 

282 

903959 

157 

872640 

439 

127360 

17 

44 

776768 

282 

903864 

157 

872903 

439 

127097 

10 

45 

776937 

282 

903770 

157 

873167 

43!) 

120833 

15 

46 

777106 

282 

903676 

157 

873430 

439 

126570 

14 

47 

777275 

281 

903581 

157 

873694 

439 

120306 

13 

48 

777444 

281 

903487 

157 

873957 

439 

120043 

12 

49 

777013 

281 

903392 

158 

874220 

439 

125780 

11 

50 

777781 

281 

903298 

158 

874484 

439 

125516 

II) 

51 

9-777950 

281 

9-903203 

158 

9-874747 

439 

10-125253 

9 

52 

778119 

281 

903108 

158 

875010 

439 

124990 

8 

53 

778287 

280 

9(0014 

158 

875273 

438 

124727 

7 

54 

778455 

280 

902919 

158 

875536 

438 

124404 

6 

55 

778(524 

280 

902824 

158 

875800 

438 

124200 

5 

50 

778792 

280 

902729 

158 

876063 

438 

123937 

4 

57 

77PSK50 

280 

902034 

KB 

876326 

438 

123(574 

3 

58 

779128 

280 

902539 

159 

870589 

438 

123411 

2 

59 

779295 

279 

902444 

159 

876851 

438 

123)49 

1 

60 

779403 

279 

903349 

159 

8771  14 

438 

122886 

0 

|      Cotang. 


I         Tang. 


53  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (37  Degrees.)      235 


M.     Sine    I   D.   |  Cosine   1   D.   I   Tan*.   |   D.   |   Cotang. 

0 

9-779463 

279 

9-902349 

159 

9-877114 

438 

10-122886 

60 

1 

779631 

279 

902253 

159 

877377 

438 

122(523 

59 

2 

779798 

279 

902158 

159 

877640 

438 

122360 

58 

3 

779966 

279 

902063 

159 

877903 

438 

122097 

57 

4 

780133 

279 

901967 

159 

878165 

438 

121835 

56 

5 

780300 

278 

901872 

159 

878428 

438 

121572 

55 

6 

780467 

278 

901776 

159 

878691 

438 

121309 

54 

7 

780634 

278 

901681 

159 

878953 

437 

121047 

53 

R 

780801 

278 

90  J  585 

159 

879216 

437 

120784 

;>2 

0 

780968 

278 

901490 

159 

879478 

437 

120522 

51 

10 

781134 

278 

901394 

160 

879741 

437 

120259 

50 

]1 

9-781301 

277 

9-901298 

160 

9-880003 

437 

10-119997 

49 

is 

781468 

277 

901202 

160 

880265 

437 

119735 

48 

13 

781634 

277 

901106 

160 

880528 

437 

119472 

47 

14 

781800 

277 

901010 

160 

880790 

437 

119210 

46 

15 

781966 

277 

900914 

160 

881052 

437 

118948 

45 

10 

782132 

277 

900818 

160 

881314 

437 

118686 

44 

IT 

782298 

276 

900722 

]60 

881573 

437 

1J8424 

4;J 

18 

782464 

276 

900626 

160 

881839 

437 

118161 

42 

11) 

782630 

276 

900529 

160 

882101 

4H7 

117899 

41 

20 

782796 

276 

900433 

161 

882363 

436 

117637 

40 

21 

9-782961 

276 

9-900337 

161 

9-882625 

436 

10-117375 

3'J 

23 

783127 

276 

900240 

161 

882887 

436 

1171J3 

38 

S3 

783292 

275 

900144 

Ifil 

883148 

436 

116852 

37 

24 

783458 

275 

900047 

Itil 

883410 

436 

116590 

36 

So 

783623 

275 

899951 

361 

8*3672 

436 

116328 

35 

96 

783788 

275 

899854 

J61 

883934 

436 

116066 

34 

27 

783953 

275 

899757 

161 

884196 

436^ 

115804 

33 

38 

784118 

275 

899660 

161 

884457 

436 

115543 

32 

2!) 

784282 

274 

899564 

161 

884719 

436 

115281 

31 

30 

784447 

274 

89941)7 

162 

884980 

436 

115020 

30 

31 

9-784612 

274 

9-899370 

162 

9-885242 

436 

10-114758 

29 

33 

784776 

274 

899273 

162 

885503 

436 

114497 

28 

33 

784941 

274 

899176 

162 

885765 

436 

114235 

27 

34 

785105 

274 

899078 

162 

886026 

43(5 

113974 

26 

35 

785269 

273 

898981 

162 

886288 

436 

113712 

25 

36 

785433 

273 

898884 

162 

886549 

435 

113451 

24 

37 

785597 

273 

898787 

162 

886810 

435 

1131;)0 

23 

38 

785761 

273 

898689 

162 

887072 

435 

112928 

22 

39 

785925 

273 

898592 

162 

887333 

435 

112667 

21 

40 

786089 

273 

898494 

163 

887594 

435 

11240G 

20 

41 

9-786252 

272 

9-898397 

163 

9-887855 

435 

10-112145 

19 

4-2 

786416 

272 

898299 

163 

888116 

435 

11J8H4 

18 

43 

786579 

272 

898202 

163 

888377 

435 

111623 

17 

44 

786742 

272 

898104 

163 

888(539 

435 

111361 

16 

4."> 

786906 

272 

898006 

163 

888'JOO 

435 

111100 

15 

4(1 

787069 

272 

897908 

163 

889160 

435 

110840 

14 

47 

787232 

271 

897810 

163 

889421 

435 

110579 

13 

48 

787395 

271 

897712 

163 

889(182 

435 

110318 

13 

4!)  ' 

787557 

271 

897614 

163 

889943 

435 

110057 

11 

50 

787720 

271 

897516 

163 

890204 

434 

109796 

10 

f>l 

9-787883 

271 

9-897418 

164 

9-890465 

434 

10-109535 

9 

52 

788045 

271 

897320 

164 

89(1725 

434 

109275 

8 

5:5 

788208 

271 

8!  (72-2-2 

164- 

890986 

434 

109014 

7 

54 

788370 

270 

8971  -23 

164 

891247 

434 

108753 

6 

55 

7*8532 

270 

897025 

164 

891507 

434 

108493 

5 

56 

7HW94 

270 

PW926 

164 

891768 

434 

108232 

4 

57 

7^'8856 

270 

896828 

Ifi4 

892028 

434 

1071)72 

3 

58 

789018 

270 

896729 

164 

892289 

434 

1077  11 

2 

30 

7891  80 

270 

896K3I 

164 

892549 

434 

107451 

1 

GO 

789342 

269 

896532 

164 

892810 

434 

107190 

0 

|       Cosine       | 


|        Sine 


I 

52  Degrees. 


Cotang. 


I        Tang. 


236       (38  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.    |         Sine        |       D.      |       Cosine       |      D.       |       Tang.       | 


Cotang.       | 


u 

9-789342 

269 

9-896532 

164 

9-892X10 

434 

10-107190 

60 

1 

789504 

269 

896433 

165 

893070 

434 

106930 

59 

2 

789665 

269 

896335 

165 

893331 

434 

Hi6GG9 

58 

3 

789827 

269 

896236 

165 

893591 

434 

100409 

57 

4 

789988 

269 

896137 

165 

893X51 

434 

106149 

56 

5 

790149 

269 

896038 

165 

894JI1 

434 

105X89 

55 

6 

790310 

268 

895939 

165 

894371 

434 

105029 

54 

7 

790471 

268 

895840 

165 

894632 

433 

105368 

53 

8 

790632 

268 

895741 

165 

894892 

433 

105108 

52 

9 

7907  y3 

2(58 

895641 

165 

895152 

433 

104X48 

51 

JO 

790954 

268 

895542 

165 

895412 

433 

104588 

50 

]] 

9-791115 

268 

9-895443 

166 

9-895672 

433 

10-104328 

49 

12 

791275 

2b7 

895343 

j(W 

895932 

433 

1040(58 

48 

J3 

79  J  436 

267 

895244 

166 

896192 

433 

103808 

47 

14 

791  596 

267 

895145 

5(5 

896452 

433 

103548 

46 

15 

791757 

267 

895U45 

]-i6 

896712 

433 

1032X8 

45 

16 

791917 

867 

894945 

]i>6 

896971 

433 

103029 

44 

17 

792077 

267 

81)4846 

]i>6 

897231 

433 

1027(59 

43 

18 

792237 

266 

894746 

166 

897491 

433 

102509 

42 

19 

792397 

266 

81)4046 

166 

897751 

433 

102249 

41 

20 

792557 

266 

89454(5 

166 

898010 

433 

101990 

40 

21 

9-792716 

266 

9-894446 

167 

9-8982TO 

433 

10-101730 

39 

22 

792876 

266 

894346 

1(57 

89X530 

433 

101470 

38 

23 

793035 

266 

894246 

167 

898789 

433 

101211 

37 

24 

793195 

265 

894146 

167 

899(1-19 

432 

100951 

36 

25 

793354 

265 

84*4046 

167 

X99.'I(IS 

432 

j  00692 

35 

2G 

793514 

265 

893946 

167 

899568 

432 

100432 

34 

27 

793073 

265 

893846 

167 

X99827 

432 

]  00  173 

33 

28 

793832 

265 

893745 

167 

900086 

432 

099914 

32 

29 

793991 

265 

893645 

167 

900346 

432 

099654 

31 

30 

794150 

264 

893544 

167 

900605 

432 

099395 

30 

31 

9-794308 

264 

9-893444 

1C8 

9-90(1864 

432 

10-099136 

29 

32 

794467 

264 

893343 

168 

901124 

432 

0!  18876 

28 

33 

794626 

264 

893243 

168 

901383 

432 

098017 

34 

794784 

264 

893142 

168 

901642 

432 

098358 

96 

35 

794942 

264 

893041 

108 

901901 

432 

098099 

25 

30 

795101 

264 

8929-10 

108 

902160 

432 

097840 

24 

37 

795259 

264 

892X39 

168 

902419 

432 

097581 

23 

38 

795417 

263 

892739 

108 

902679 

432 

097321 

22 

39 

795575 

263 

892638 

168 

902938 

432 

097062 

21 

40 

795733 

263 

892536 

168 

903197 

431 

096803 

20 

41 

9-795891 

263 

9-892435 

169 

9-903455 

431 

10-096545 

19 

42 

796049 

263 

892334 

169 

903714 

431 

096286 

18 

43 

796206 

263 

892233 

169 

903973 

431 

09(5027 

17 

14 

790364 

262 

892132 

169 

904232 

431 

095768 

10 

45 

796521 

262 

892(130 

169 

904491 

431 

09550!) 

15 

46 

79(5679 

262 

891929 

1(59 

904750 

431 

095250 

14 

47 

79(5X36 

262 

891X27 

109 

905008 

431 

09-4  9!  12 

13 

48 

796993 

262 

891726 

109 

905207 

431 

094733 

12 

49 

797150 

261 

891624 

]09 

905526 

431 

094474 

11 

50 

797307 

261 

891523 

170 

905784 

431 

094216 

10 

51 

9-797464 

261 

9-891421 

70 

9-906043 

431 

10-093957 

9 

52 

797621 

261 

891319 

70 

900302 

431 

093098 

8 

53 

797777 

261 

891217 

70 

906500 

431 

093440 

7 

54 

791934 

261 

891115 

70 

906X19 

431 

093181 

6 

55 

79X091 

261  ' 

891013 

70 

907077 

431 

0925)23 

5 

56 

79X247 

261 

890911 

70 

907336 

431 

092664 

4 

57 

798403 

260 

890809 

70 

907594 

431 

092406 

3 

58 

79X5HO 

2(50 

890707 

170 

907X52 

431 

092148 

2 

SB 

798716 

2(50 

890605 

170 

90X111 

430 

Oil  1X89 

1 

60 

798872 

260 

890503 

170 

908369 

430 

091631 

0 

Conne      | 


Sine       | 


|      Cotang.     | 


Tang. 


51  Degrees. 


LOGARITHMIC  SINES,  COSINES,  ETC.     (39  Degrees.)      237 


M.    |         Sine        I 


|         D.       |       Tang.        |        I).       |      Coin 


0 

9-798872 

260 

9-890503 

170 

9-908309 

430 

10-091631 

60 

1 

799028 

260 

890400 

171 

908G28 

430 

091372 

59 

2 

799184 

260 

891)298 

171 

908886 

430 

091114 

58 

3 

791)339 

259 

89J195 

71 

909144 

430 

090856 

57 

4 

79949.1 

259 

890093 

71 

909402 

430 

090598 

56 

5 

799(551 

259 

88.)9;)0 

71 

909660 

430 

090340 

55 

6 

79980G 

o-,(j 

889888 

71 

909918 

430 

090082 

54 

7 

799MB 

259 

889785 

71 

910177 

430 

089823 

53 

8 

800117 

259 

889682 

71 

910435 

430 

089565 

52 

9 

800272 

258 

889579 

71 

910693 

430 

089307 

51 

10 

800427 

258 

889477  k 

71 

910951 

430 

089049 

50 

11 

9-800582 

258 

9-889374 

72 

9-911209 

430 

10-088791 

49 

12 

800737 

258 

889271 

72 

91  1467 

430 

088533 

48 

13 

800892 

258 

8891(58 

72 

91  1724 

430 

088276 

47 

14 

801047 

258 

889064 

72 

911982 

430 

088018 

46 

IS 

801201 

258 

888961 

72 

912240 

430 

0877(50 

45 

16 

801  350 

257 

888858 

72 

912498 

430 

087502 

44 

17 

801511 

257 

888755 

72 

912756 

430 

087244 

43 

18 

801665 

257 

888651 

72 

913014 

429 

086986 

42 

.1!) 

801819 

257 

888548 

72 

913271 

429 

086729 

41 

20 

801973 

257 

888444 

73 

913529 

429 

086471 

40 

21 

9-802128 

257 

9-888341 

73 

9-913787 

429 

10-086213 

39 

2-2 

802282 

256 

888237 

73 

914044 

429 

085956 

38 

2i{ 

802436 

256 

888134 

73 

914302 

429 

085(598 

37 

24 

802589 

256 

888030 

73 

914560 

429 

085440 

36 

2.-, 

802743 

256 

887926 

73 

91481*7 

429 

085183 

35 

2<i 

802897 

256 

887822 

73 

915075 

429 

084925 

34 

27 

803050 

256 

887718 

73 

915332 

429 

084668 

33 

28 

803204 

256 

887614 

73 

915590 

429 

084410 

32 

21) 

803357 

255 

887510 

73 

915847 

429 

084153 

31 

30 

803511 

255 

887406 

74 

916104 

429 

083896 

30 

31 

9-803664 

255 

9-887302 

74 

9-916362 

429 

10-083638 

29 

32 

803817 

255 

887198 

x  ~4 

916619 

429 

083381 

28 

33 

803970 

255 

887093 

74 

91(5877 

429 

083  J2J 

27 

34 

804123 

255 

886989 

74 

917134 

429 

082866 

26 

3:, 

804276 

254 

886885 

74 

917391 

429 

082(509 

25 

3U 

804-128 

254 

886780 

74 

917648 

429 

082352 

24 

37 

804581 

254 

886676 

74 

917905 

429 

082095 

23 

38 

804734 

254 

886571 

74 

918163 

428 

081837 

22 

39 

804886 

254 

886466 

74 

9184>20 

428 

081580 

21 

40 

805039 

254 

886362 

75 

918677 

428 

081323 

20 

41 

9-805191 

254 

9-886257 

75 

9-918934 

428 

10-081066 

19 

42 

805343 

253' 

886152 

75 

919191 

428 

0808i)9 

18 

43 

805495 

253 

88(5047 

75 

919448 

428 

080552 

17 

44 

805647 

253 

885942 

75 

919705 

428 

080295 

16 

45 

805799 

253 

885837 

75 

919962 

428 

080038 

15 

46 

805951 

253 

885732 

75 

920219 

428 

079781 

14 

47 

80(5103 

253 

885(527 

75 

92047(5 

428 

079524 

13 

4H 

806254 

253 

885522 

75 

920733 

428 

0792(57 

12 

4!) 

801  140(5 

252 

885416 

75 

920990 

428 

079010 

11 

50 

806557 

252 

885311 

76 

921247 

428 

078753 

10 

51 

9-80(5709 

252 

9-885205 

76 

9-921503 

428 

10'()78497 

9 

52 

80(5860 

252 

885100 

76 

9217(50 

428 

078240 

8 

5:5 

807011 

252 

884994 

76 

922017 

428 

077983 

7 

54 

807163 

252 

884889 

76 

922274 

428 

077726 

6 

55 

807314 

252 

884783 

76 

922530 

428 

077470 

5 

50 

807465 

251 

884677 

76 

922787 

428 

077213 

4 

57 

807615 

251 

884572 

76 

923044 

428 

076956 

3 

58 

H07766 

251 

8844(56 

76 

923300 

428 

07(5700 

2 

59 

807917 

251 

884360 

176 

923557 

427 

876443 

1 

GO 

808067 

251 

884254 

177 

923813 

427 

076187 

0 

Cosine      | 


I. 

50  Decrees. 


Tang. 


238       (40  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M. 

Sine 

D: 

Cosine   | 

D.   | 

Tang. 

D. 

C«anff.   | 

0 

9-808007 

251 

9-884254 

177 

9-923813 

427 

10-070187 

1 

8118218 

251 

884148 

177 

924070 

427 

075930 

2 

8083(58 

251 

884042 

177 

924327 

427 

075073 

3 

808519 

250 

883936 

177 

924583 

427 

0754J7 

4 

808609 

250  , 

883829 

177 

924840 

427 

0751(50 

5 

808819 

250 

883723 

177 

925096 

427 

074904 

6 

808909 

250 

883017 

177 

925352 

427 

074(548 

7 

8091  19 

250 

883510 

177 

925009 

427 

074391 

8 

809-209 

250 

883404 

177 

9258(55 

427 

074  J  35 

9 

8(19419 

249 

883297 

78 

926122 

427 

073878 

JO 

809509 

249 

883191 

78 

920378 

427 

073022 

11 

9-809718 

249 

9-883084 

78 

9-9200:14 

427 

10-073366 

12 

809868 

249 

8821*77 

78 

920890 

427 

073110 

13 

810017 

249 

882871 

78 

927147 

427 

07-2853 

J4 

810167 

249 

882764 

78 

927403 

427 

072597 

15 

810316 

248 

882657 

78 

927059 

427 

072341 

16 

810465 

248 

882550 

78 

927915 

427 

072085 

17 

8i0614 

248 

882443 

78 

928171 

427 

071829 

18 

810763 

248 

882336 

79 

928427 

427 

071573 

19 

810912 

248 

882229 

79 

928(583 

427 

071317 

2U 

81  J  061 

248 

882121 

79 

928940 

427 

071060 

21 

9-811210 

248 

9-882014 

79 

9-929196 

427 

10-070804 

22 

811358 

247 

88  J  907 

79 

929452 

427 

070548 

23 

811507 

247 

881799 

79 

929708 

427 

070292 

24 

81  1055 

247 

8*1092 

79 

929904 

426 

070036 

25 

811804 

247 

881584- 

79 

930220 

426 

009780 

26 

811952 

247 

881477 

79 

930475 

426 

069525 

27 

812100 

247 

881309 

79 

930731 

426 

0(59209 

23 

8  12248 

247 

881201 

180 

930987 

426 

069013 

29 

812396 

246 

881153 

180 

931243 

426 

008757 

30 

812544 

246 

881046 

180 

931499 

426 

068501 

31 

9-812092 

246 

9-880938 

180 

9-931755 

426 

10-068245 

32 

812H40 

246 

880830 

180 

932010 

426 

OG7990 

33 

812988 

246 

880722 

180- 

9322GG 

426 

OG7734 

34 

813135 

246 

880013 

180 

QOOjOO 

426 

CG7478 

35 

813283 

246 

880505 

180 

932778 

426 

OC7222 

36 

813430 

245 

880397 

180 

933033 

426 

06G9G7 

37 

813578 

245 

88(1289 

181 

933289 

426 

OGG711 

38 

813725 

245 

880180 

181 

933545 

426 

OGG455 

39 

813872 

245 

88W072 

181 

933800 

426 

06G200 

40 

814019 

245 

879963 

181 

934056 

426 

065944 

41 

9-814166 

245 

9-879855 

181 

9-934311 

426 

10-065689 

42 

814313 

245 

87'.)746 

181 

934507 

426 

065433 

43 

814400 

244 

879037 

181 

934823 

426 

065177 

44 

814(507 

244 

879529 

181 

935078 

426 

004922 

45 

814753 

244 

879420 

181 

935333 

420 

0(540(57 

46 

8I49IK) 

244 

87!  13  11 

181 

935589 

426 

0644  1  1 

47 

815046 

244 

879202 

182 

935844 

426 

064156 

48 

815193 

244 

879093 

182 

936100 

4-2(5 

0(53900 

4!) 

HI  5339 

244 

878984 

182 

930355 

4-26 

003045 

50 

815485 

243 

878875 

182 

930(510 

426 

063390 

51 

9-815031 

243 

9-878766 

182 

9-930806 

425 

10-003134 

52 

815778 

243 

878056 

182 

937  121 

425 

002879 

53 

8  15!  124 

243 

878547 

182 

937376 

425 

00-2024 

54 

H|  til  nil) 

243 

878438 

182 

937(532 

425 

002308 

55 

810215 

243 

878328 

182 

937887 

425 

002  113 

56 

810301 

243 

8782  19 

183 

938142 

425 

001858 

57 

81  651  17 

24-2 

878109 

183 

938398 

425 

001002 

58 

810052 

242 

877999 

183 

938053 

425 

001347 

59 

816798 

242 

877890 

183 

938908 

425 

001092 

60 

810943 

242 

877780 

183 

939163 

425 

0(50837 

I       Cosine 


Sine       |  |      Cotang.      | 

49  Degrees. 


Tang.       I   M. 


LOGARITHMIC  SINES,  COSINES,  ETC.    (41  Degrees.)      239 


Sine [       D. | 


Tang. 


D.       1     Cotang. 


0 

9-816943 

242 

9  877780 

183 

9-939163 

425 

10-060837 

60 

1 

817088 

242 

877670 

183 

9394  J  8 

425 

060582 

59 

2 

817233 

242 

8775t>0 

183 

939673 

425 

060327 

58 

3 

817379 

242 

877450 

183 

939928 

425 

060072 

57 

4 

817524 

241 

877340 

1H3 

941JJ83 

425 

059F17 

56 

5 

817668 

241 

877230 

184 

9404:i8 

425 

059562 

55 

6 

817813 

241 

877120 

184 

940694 

425 

059306 

54 

7 

817958 

241 

877010 

184 

940949 

425 

059051 

53 

8 

818103 

241 

871)899 

184 

941204 

425 

058796 

52 

9 

818247 

241 

876789 

184 

941458 

425 

058542 

51 

10 

818392 

241 

876678 

184 

941714 

425 

058286 

50 

11 

9-818536 

240 

9  876568 

184 

9-941968 

425 

10-058032 

49 

12 

818681 

240 

8715457 

184 

942223 

425 

057777 

48 

is 

818825 

240 

876347 

184 

942478 

425 

057522 

47 

14 

818969 

240 

876236 

185 

942733 

425 

057267 

46 

J5 

819113 

240 

876125 

185 

942988 

425 

057012 

45 

J6 

819257 

240 

87(5014 

185 

943243 

425 

056757 

44 

17 

819401 

240 

875904 

185 

943498 

425 

056502 

43 

J8 

819545 

239 

875793 

185 

943752 

425 

056248 

42 

19 

819689 

239 

875682 

185 

944007 

425 

055993 

41 

20 

819832 

239 

875571 

185 

944262 

425 

055738 

40 

21 

9-819976 

239 

9-875459 

185 

9-944517 

425 

10-055483 

39 

22 

820120 

239 

875348 

185 

944771 

424 

055229 

38 

23 

820263 

239 

875237 

185 

945026 

424 

054974 

37 

24 

820406 

239 

875126 

186 

945281 

424 

054719 

36 

25 

820550 

238 

875014 

186 

945535 

424 

054465 

35 

26 

820693 

238 

874903 

186 

945790 

424 

054210 

34 

27 

820836 

238 

874791 

186 

946045 

424 

053955 

33 

28 

820979 

238 

874680 

186 

946299 

424 

053701 

32 

29 

821122 

238 

874568 

186 

946554 

424 

053446 

31 

30 

821265 

238 

874456 

186 

946808 

424 

053192 

30 

31 

9-821407 

238 

9-874344 

186 

9-947063 

424 

10-052937 

29 

32 

821550 

238 

874232 

187 

947318 

424 

052(582 

28 

33 

82  1093 

237 

874121 

187 

947572 

424 

052428 

27 

34 

821835 

237 

874009 

187 

947826 

424 

052174 

2e 

35 

821977 

237 

873896 

187 

948081 

424 

051919 

25 

30 

822120 

237 

873784 

187 

948336 

424 

051664 

24 

37 

8222G2 

237 

873672 

187 

948590 

424 

051410 

23 

38 

822404 

237 

8735CO 

187 

948844 

424 

051156 

22 

39 

822546 

237 

873448 

187 

949099 

424 

050901 

21 

40 

822688 

236 

873335 

187 

949353 

424 

050647 

20 

41 

9-822830 

236 

9-873223 

187 

9-949607 

424 

10.050393 

19 

42 

822972 

236 

873110 

188 

949862 

424 

050138 

18 

43 

823114 

236 

872998 

188 

950116 

424 

049884 

17 

44 

823255 

236 

872885 

188 

950370 

424 

049630 

16 

45 

823397 

236 

872772 

188 

950625 

424 

049375 

15 

40 

823539 

236 

872659 

188 

950879 

424 

049121 

14 

47 

823680 

235 

872547 

188 

951133 

424 

0488(57 

13 

48 

823821 

235 

872434 

188 

951388 

424 

048612 

12 

49 

823963 

235 

872321 

188 

951642 

424 

048358 

11 

50 

824104 

235 

872208 

188 

951896 

424 

048104 

10 

51 

9-824245 

235 

9-872095 

189 

9-9521.50 

424 

10-047850 

9 

52 

824:186 

235 

871981 

189 

952405 

424 

047595 

3 

53 

82-4527 

235 

871  HIM 

189 

952659 

424 

047341 

7 

54 

8246(>8 

234 

871755 

189 

952913 

424 

047087 

6 

55 

824808 

«:J4 

871641 

189 

9531(57 

423 

04H833 

5 

56 

824949 

2M 

871528 

189 

953421 

423 

046579 

4 

57 

825090 

234 

871414 

189 

953675 

423 

046325 

3 

58 

825230 

234 

871301 

189 

953929 

423 

046071 

2 

59 

825371 

£14 

871187 

189 

954183 

423 

045817 

60 

825511 

234 

871073 

190 

954437 

423 

045563 

0 

Sine         |  |       Cotang. 

48  Degrees. 


I        Tang.       f 


240       (42  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


|       D. 


|        Tang. 


D.        |       Cotang. 


0 

9-825511 

234 

9-871073 

190 

9-954437 

423 

10-045563 

(50 

1 

825651 

233 

870960 

190 

954691 

423 

045309 

59 

2 

8-25791 

233 

870846 

190 

954945 

423 

045055 

58 

3 

8-25J31 

233 

870732 

190 

955200 

423 

044800 

57 

4 

82<>G7l 

233 

870618 

190 

955454 

423 

044546 

56 

5 

82(5211 

233 

870504 

190 

955707 

423 

044-293 

55 

6 

826351 

233 

870390 

1'JO 

955S«)1 

423 

044039 

5-1 

7 

82(5491 

233 

870276 

190 

956215 

423 

043785 

53 

8 

826631 

233 

870161 

190 

956469 

423 

043531 

52 

9 

826770 

232 

870047 

191 

956723 

423 

043-277 

51 

10 

826910 

232 

869933 

191 

95(5977 

423 

043023 

50 

11 

9-827049 

232 

9-869818 

191 

9-957231 

423 

10-042769 

4'.) 

12 

827  189 

232 

869704 

19L 

957485 

423 

042515 

48 

13 

8273-28 

232 

869589 

191 

957739 

423 

04i>261 

47 

14 

827467 

232 

8(59474 

191 

9579SJ3 

423 

042007 

4(5 

15 

827606 

232 

869300 

191 

958246 

423 

041754 

45 

16 

827745 

232 

869245 

191 

958500 

423 

041500 

44 

17 

827884 

231 

869130 

191 

958754 

423 

041246 

43 

18 

828023 

231 

8(59015 

192 

959008 

423 

040992 

42 

19 

828162 

231 

8681)00 

192 

9592(52 

423 

0407:18 

41 

20 

828301 

231 

868785 

192 

959516 

423 

040484 

40 

21 

9-828439 

231 

9-86F670 

192 

9-959769 

423 

10-040231 

39 

22 

828578 

231 

868555 

192 

960023 

423 

039977 

3rt 

23 

828716 

231 

868440 

192 

960277 

423 

039723 

37 

24 

828855 

230 

868324 

192 

960531 

423 

039469 

3(5 

25 

828993 

230 

8(58209 

192 

960784 

423 

039216 

35 

26 

829131 

230 

8(58093 

192 

961038 

423 

038962 

34 

27 

829269 

230 

8(57978 

193 

961291 

423 

038709 

33 

28 

829407 

230 

8678(52 

193 

961545 

423 

0-8455 

32 

29 

829545 

230 

867747 

193 

961799 

423 

038201 

31 

30 

829(583 

230 

867631 

193 

962052 

423 

037948 

30 

31 

9-829821 

229 

9-867515 

193 

9-962306 

423 

10-037C94 

29 

32 

829959 

229 

867399 

193 

962560 

423 

037440 

28 

33 

830097 

229 

867283 

193 

9C2813 

423 

037187 

27 

34 

830234 

229 

867167 

193 

963067 

423 

036933 

26 

35 

830372 

229 

867051 

193 

963320 

423 

0366PO 

25 

36 

830509 

229 

86(5935 

194 

963574 

423 

036426 

24 

37 

&30646 

229 

866819 

194 

9C3827 

423 

036173 

23 

38 

830784 

229 

8(i(i703 

194 

9f:4081 

423 

035919 

22 

39 

830921 

228 

8(5(5586 

194 

964335 

423 

035665 

21 

40 

831058 

228 

866470 

194 

964588 

422 

035412 

20 

41 

9-831195 

228 

9-866353 

194 

9-964842 

422 

10-035158 

19 

42 

831332 

228 

86(5237 

194 

965095 

422 

034905 

18 

43 

831469 

228 

866120 

194 

965349 

422 

034651 

17 

44 

831(506 

228 

866004 

195 

965(502 

422 

034398 

16 

45 

831742 

228 

865887 

195 

965855 

422 

034145 

15 

46 

831879 

228 

865770 

195 

966109 

422 

033891 

14 

47 

832015 

227 

8G5C53 

195 

966362 

422 

033638 

13 

48 

832152 

227 

805536 

195 

966616 

422 

033384 

12 

49 

832288 

227 

805-1  19 

195 

96(5869 

422 

033131 

11 

50 

832425 

227 

8(55302 

195 

9(571  '23 

422 

032877 

10 

51 

9-832561 

227 

9'805IF5 

195 

9-967376 

422 

10-032624 

9 

52 

832(597 

227 

865068 

195 

9(57(529 

422 

032371 

8 

53 

832833 

227 

8649.50 

195 

9C.7W3 

422 

032117 

7 

54 

832969 

22f> 

864833 

196 

968136 

422 

031864 

6 

55 

833105 

226 

864716 

196 

968389 

422 

03  1C-  11 

5 

56 

83:;24i 

226 

8(54598 

196 

9(58.643 

422 

031357 

4 

57 

833377 

226 

8(54481 

196 

9(58896 

422 

031104 

3 

58 

833512 

226 

864303 

196 

969149 

422 

030H51 

2 

59 

83IH548 

226 

864245 

196 

969403 

422 

030597 

1 

60 

833783 

226 

8(54127 

196 

969(556 

422 

030344 

0 

|      Cotang. 


I         Tang.      |  M. 


47  Degrees. 


LOGARITHMIC  SINLS,  COSINES,  ETC.    (43  Degrees.)      241 


Cos.nc        |       D.       | 


|       D. 


0 

9-833783 

226 

9-864127 

196 

9-969056 

4-J2 

10-030344  |  60 

1 

833919 

225 

864010 

1% 

969909 

422 

030091 

59 

2 

834054 

225 

86381(2 

197 

970162 

422 

029838 

58 

3 

834189 

225 

803774 

197 

970416 

422 

029584 

57 

4 

834325 

2-25 

863656 

197 

970669 

422 

029331 

56 

5 

834460 

225 

863538 

197 

970922 

422 

029078 

55 

6 

834595 

225 

863419 

11)7 

971175 

422 

02H825 

54 

7 

834730 

225 

803301 

197 

971429 

422 

028571 

53 

8 

834865 

225 

863J83 

197 

971682 

422 

028318 

52 

9 

834999 

2-24 

863064 

197 

971935 

422 

028065 

51 

10 

835J34 

224 

8G2946 

198 

972188 

422 

027812 

50 

11 

9-835269 

224 

9-862827 

198 

9-972441 

422 

10-027559 

49 

12 

835403 

224 

862709 

198 

972694 

422 

027306 

48 

13 

835538 

224 

862590 

198 

972948 

422 

027052 

47 

14 

835672 

224 

862471 

198 

973201 

422 

026799 

46 

15 

835807 

224 

862353 

198 

973454 

422 

026546 

45 

16 

835941 

224 

862234 

198 

*  973707 

422 

026293 

44 

17 

&36075 

223 

862115 

198 

973960 

422 

026040 

43 

18 

836209 

223 

861996 

198 

974213 

422 

025787 

42 

19 

836343 

223 

861877 

198 

974466 

422 

025534 

41 

20 

836477 

223 

861758 

199 

974719 

422 

025281 

40 

21 

9-836611 

223 

9-861638 

199 

9-974973 

422 

10-025027 

39 

22 

836745 

2-23 

861519 

199 

975226 

422 

024774 

38 

23 

836878 

2-23 

861400 

199 

975479 

422 

024521 

37 

24 

837012 

222 

801280 

199 

975732 

422 

024268 

36 

25 

837146 

ooo 

861161 

199 

975985 

422 

024015 

35 

26 

837279 

222 

861041 

199 

976238 

422 

023762 

34 

27 

837412 

222 

800i>22 

199 

976491 

422 

023509 

33 

28 

837546 

222 

860802 

199 

976744 

422 

023256 

32 

i2i) 

837679 

222 

860682 

200 

976997 

422 

023003 

31 

30 

837812 

222 

800562 

200  . 

977250 

422 

022750 

30 

31 

9-837945 

222 

9-860442 

200 

9-977503 

422 

10-022497 

29 

32 

838078 

221 

800322 

200 

977756 

422 

022244 

28 

33 

838211 

221 

860202 

200 

978009 

422 

021991 

27 

34 

838344 

221 

860082 

200 

978202 

422 

021738 

26 

35 

838477 

221 

859962 

200 

978515 

422 

021485 

25 

30 

838010 

221 

859842 

200 

978768 

422 

021232 

24 

37 

838742 

221 

859721 

201 

979021 

422 

020979 

23 

38 

838875 

221 

859601 

201 

979274 

422 

020726 

22 

39 

839007 

221 

859480 

201 

979527 

422 

020473 

21 

40 

839140 

220 

859360 

201 

979780 

422 

020220 

20 

41 

9-839272 

220 

9-859239 

201 

9-980033 

422 

10-019967 

19 

42 

839404 

220 

859119 

201 

980280 

422 

019714  ' 

18 

43 

839536 

220 

858998 

201 

980538 

422 

019462 

17 

44 

839668 

220 

858877 

201 

980791 

421 

019209 

16 

45 

839800 

220 

858756 

202 

981044 

421 

018956 

15 

46 

839932 

220 

858635 

202 

981297 

421 

018703 

14 

47 

840064 

219 

858514 

202 

981550 

421 

018450 

13 

48 

840196 

219 

858393 

202 

981803 

421 

018197 

12 

49 

840328 

2J9 

858272 

202 

9H2056 

421 

(117944 

11 

50 

840459 

219 

858151 

202 

982309 

421 

017691 

10 

51 

9-840591 

219 

9-858029 

202 

9-982562 

421 

10-017438 

9 

52 

840722 

219 

857908 

202 

982814 

421 

017186 

8 

53 

840854 

219 

857786 

202 

983067 

421 

016933 

7 

54 

840985 

219 

857665 

203 

983320 

421 

010080 

6 

55 

841116 

218 

857543 

203 

983573 

421 

016127 

5 

56 

841247 

218 

857422 

203 

98.-J826 

421 

016174 

4 

57 

841378 

218 

857300 

203 

984079 

421 

015921 

3 

58 

841509 

218 

857178 

203 

984331 

42i 

015669 

2 

59 

841640 

218 

857056 

203 

984584 

421 

015416 

1 

60 

841771 

218 

856934 

203 

984837 

421 

015163- 

0 

|  I       Cotang. 

46  Degrees. 


J          Tang.       I 


242       (44  Degrees.)     LOGARITHMIC  SINES,  COSINES,  ETC. 


M.     Sine      JD   1  Cosine      D.      Tan?.    |   D.      Cotnn?.   | 

0 

9-841771 

•218 

9-850934 

203 

9-984837 

421 

10-015103 

tin 

1 

8411)02 

218 

850812 

203 

985090 

4-21 

014910 

59 

2 

84-2033 

218 

850090 

204 

985343 

421 

OJ4057 

58 

3 

842  1G3 

217 

856508 

204 

985596 

421 

014404 

57 

4 

842294 

217 

856446 

204 

985848 

421 

01  1152 

5i> 

5 

8424-24 

217 

850323 

204 

980101 

421 

013899 

55 

(i 

842555 

217 

850201 

204 

980354 

421 

013040 

54 

7 

842085 

217 

850078 

204 

980007 

4-21 

01  3393 

53 

B 

842815 

217 

8551)50 

204 

986800 

421 

013140 

52 

9 

842940 

217 

855833 

204 

987112 

421 

012888 

51 

10 

84307G 

217 

855711 

205 

987305 

421 

012035 

50 

11 

9-843206 

216 

9-855588 

205 

9-987018 

421 

10-012382 

4!> 

12 

843330 

210 

855405 

205 

987871 

421 

012129 

48 

13 

843406 

216 

85534-2 

205 

9881-23 

421 

011877 

47 

14 

84351)5 

216 

855219 

205 

988376 

421 

V)  11  024 

46 

15 

8437-25 

216 

855090 

205 

988029 

421 

011371 

45 

16 

843855 

216 

854973 

'  205 

988882 

421 

011118 

44 

17 

843984 

216 

854850 

205 

989134 

421 

010800 

43 

1H 

844114 

215 

854727 

200 

989387 

421 

010013 

42 

19 

844243 

215 

854603 

206 

989640 

421 

010300 

41 

20 

844372 

215 

854480 

200 

989893 

421 

010107 

40 

21 

9-844502 

215 

9-854356 

206 

9-990145 

421 

10-009855 

3!) 

22 

844031 

215 

854233 

206 

990398 

421 

009002 

an 

2!1 

844700 

215 

854109 

206 

990051 

421 

009349 

37 

24 

844889  - 

215 

8531)80 

206 

91)01)03 

421 

009097 

30 

25 

845018 

215 

853802 

200 

991156 

421 

008844 

35 

90 

845147 

215 

853738 

200 

991409 

421 

008591 

34 

27 

845270 

214 

853014 

207 

991002 

421 

008338 

33 

28 

845405 

214 

853490 

207 

91)1914 

421 

008080 

33 

2J 

845.333 

214 

8533(50 

207 

99-2107 

421 

007833 

31 

30 

845002 

214 

853242 

2J7 

99-2420 

421 

007580 

30 

31 

9-845790 

214 

9-853118 

207 

9-992072 

421 

10-007328 

29 

3-2 

845919 

214 

852994 

207 

9:i2J-25 

421 

007075 

28 

3:$ 

840047 

214 

852869 

207 

91)3178 

421 

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27 

34 

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214 

852745 

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421 

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215 

35 

840304 

214 

852020 

207 

993(583 

421 

000317 

25 

3(i 

840432 

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85-2490 

208 

99IW3G 

421 

000004 

24 

37 

840500 

213 

852371 

208 

994189 

421 

005811 

23 

38 

84IK588 

213 

852247 

208 

994441 

421 

005559 

22 

39 

840816 

213 

852122 

208 

994694 

421 

005306 

21 

40 

846944 

213 

851997 

208 

994947 

421 

005053 

20 

41 

9-847071 

213 

9-851872 

208 

9-995199 

421 

10-004801 

19 

42 

847199 

213 

851747 

208 

995452 

421 

004548 

18 

43 

8473-27 

213 

851022 

208 

995705 

421 

004295 

17 

44 

847454 

212 

851497 

209 

995957 

421 

004043 

10 

45 

847582 

212 

851372 

209 

990210 

421 

003790 

15 

46 

847709 

212 

851246 

209 

99«>403 

421 

003537 

14 

47 

847836 

212 

85  11  '21 

209 

990715 

421 

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13 

48 

847964 

212 

850996 

209 

990908 

421 

003032 

12 

41) 

848091 

212 

850870 

209 

997221 

421 

002779 

11 

50 

848218 

212 

850745 

209 

997473 

421 

002527 

10 

51 

9-848345 

212 

9-850619 

209 

9-9977-20 

421 

10-002274 

9 

53 

848472 

211 

850493 

210 

997979 

421 

009021 

8 

53 

848599 

211 

85(1308 

210 

9982:11 

421 

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7 

54 

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21J 

85024-2 

210 

998484 

421 

001516 

6 

55 

848852 

211 

850110 

210 

91IH737 

421 

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5 

50 

848979 

211 

84991)0 

210 

91)8;»81) 

421 

001011 

4 

57 

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211 

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210 

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421 

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3 

58 

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211 

849738 

210 

999495 

421 

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2 

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210 

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421 

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1 

(M) 

849485 

211 

849485 

210 

jo-oooooo 

421 

000000 

0 

Cosuw 


Ccuug. 


I         Tanp.       | 


4.5  De-Tees. 


TABLE   XIV. 
NATUKAL  SINES  AND  TANGENTS. 


246 


NATURAL   SINES. 


/ 

16° 

17a  1  18° 

19° 

20° 

21° 

22° 

28° 

/ 

0 

275  C374 

292  3717 

309  0170 

325  5C82 

342  0201 

358  3C79 

374  C066  -390  731V  60. 

1 

9170 

C499 

2936 

8432 

2935 

6395 

8763    S9S9  59 

a 

276  1915 

9280 

5702 

326  1182 

5CG8 

9110 

375  1459  .391  2166  58 

3 

4761 

93  2001 

84G8 

3932 

8400 

359  1825 

4156    5343  57 

4 

7551 

4842 

310  1234 

6681 

o43  1133 

4540 

6852    8019  56 

6 

277  0352 

7623 

3999 

9430 

38C5 

7254 

9547  -392  OG951  55 

6 

3147 

294  0403 

67  C4 

327  2179 

6597 

9968-8762243,   3371  54 

7 

5941 

3183 

9529 

4928 

9329 

3602G82    40381    C047  53 

S 

S73C 

5963 

311  2294 

767G 

3442060 

5395;    76321    8722  52 

y 

278  1530 

8743 

5058 

328.0424 

4791 

8108J 

377  0327 

393  1397  (  51 

IQ 

4324 

295  1522 

7822 

3172 

7521 

361  OS21 

3021 

4071  50 

11 

7118 

4302 

312Q586 

5919 

345  0252 

3534 

5714 

6745  49 

VI 

9911 

7081 

3349 

8666 

2982 

6246 

8408    94191  48 

13 

279  2704 

9859 

6112 

329  1413 

5712 

8958  3781101-3942093!  47 

14 

5497 

2962638 

8875 

4160 

8441 

362  1G69 

3794J    47661  46 

la 

8290 

5416 

313  1C3S 

6906 

3461171 

4380 

6486    7439!  45 

16 

280  1083 

8194 

4400 

9653 

3900 

7091 

9178  -395  01111  44 

17 

3875 

297  0971 

71GS 

330  2398 

6628 

9802 

3791870    2783!  43 

IS 

6667 

3749 

9925 

5144 

9357 

363  2512 

4562    5455  42 

18 

9459 

6526 

314  2681 

7889 

347  2085 

5222 

7253    8127 

41 

20 

281  2251 

9303 

544> 

331  0634 

4812 

7932 

9944 

3960798 

40 

21 

5042 

298  2079 

8209 

3379 

7540 

3640641 

380  21,34 

34G8 

39 

22 

7833 

4856 

315  09G9 

6123 

348  02C7 

3351 

5324 

6139 

38 

2;', 

2820624 

7632 

3730 

8867 

2994 

6059 

8014 

8809 

37 

24 

3415 

299  0408 

6490 

•3321611 

5720 

8768 

381  0704 

397  1479 

36 

25 

6205 

3184 

9250 

4355 

8447 

365  1476 

3393 

4148 

35 

26 

8995 

5959 

316  2010 

7098 

349  1173 

4184 

6082 

6818 

34 

27 

2831785 

8734 

4770 

9841 

3898 

tS91 

8770 

9486 

33 

28 

4575 

300  1509 

7529 

•3332584 

6624 

9599 

382  1459 

398  2155 

32 

29 

7364 

4284 

317  0288 

5326 

9349 

366  2306 

4147 

4823 

31 

GO 

•2840153 

7058 

3047 

80G9 

350  2074 

5012 

6834 

7491 

30 

31 

2942 

9832 

5805 

•3340810 

4798 

7719 

9522 

.3990158 

29 

32 

5731 

301  2GOC 

8563 

3552 

7523 

367  0425 

•383  2209 

2825 

28 

83 

8520 

5380 

318  1321 

6293 

351  0246 

3130 

4895 

5492 

27 

34 

•285  1308 

8153 

4079 

9034 

2970 

5836 

7582 

8158 

26 

35 

409G 

3020926 

683C 

•3351775 

5693 

8541 

•3840268 

•400  0825 

25 

86 

6884 

3699 

9593 

45U 

8416 

368  1246 

2953 

3490 

24 

37 

9671 

6471 

•319  2350 

725C 

3521139 

3950 

5t39 

6ft6 

23 

3s 

•2862458 

9244 

510C 

9996 

38G2 

6654 

8324 

8821 

22 

39 

5246 

•303  201C 

7863 

•3362735 

6584 

935S 

•385  1008 

•4011486 

21 

40 

8032 

4788 

•3200619 

5475 

930C 

•369  2061 

3C93 

4150 

20 

41 

•287  0819 

-  7559 

3374 

8214 

•853  202" 

4765    6377 

6814 

19 

42 

3605 

•3040331 

6130 

•337  0953 

4748 

7468 

906C 

9478 

18 

4:5 

6391 

3102 

8885 

3691 

7469 

•370  0170 

•3861744 

•4022141 

17 

44 

917" 

5872 

•321  1640 

6429 

•3540190 

*.S*  1 

4427 

4804 

16 

45 

•288l96o 

8G43 

439f 

916" 

2910 

5574 

7110 

7467 

15 

46 

474 

•305  1413 

7149 

•338  190o 

5630 

8276 

9792 

.4030129 

14 

47 

753 

4183 

9903 

464'. 

8350 

•371  0977 

•387  2474 

2791 

13 

48 

•289031 

6953 

•3222657 

7379 

•355  1070 

3G78 

5156 

5453 

12 

49 

310 

9723 

5411 

•339  Oil 

3789 

6379 

7837 

8114 

11 

50 

588 

•306  249L 

8164 

285 

650 

907S 

•388  0518 

•4040775 

10 

51 

867 

5261 

•3230917 

558 

922 

•3721780 

3199 

3436 

9 

52 

•290  145 

803C 

367C 

832 

•356194 

447S 

5880 

C096 

8 

53 

423 

•307  079L 

6422 

•340  106 

4C6 

717S 

8560 

8756 

7 

54 

702 

35C 

9114 

379 

738 

987* 

•389  124C 

•405  1416 

6 

55 

980 

6334 

•324192C 

653 

•357  009 

•373  2577 

391S 

4075 

5 

5f, 

•291  258 

910 

4G7£ 

926 

281 

527£ 

659S 

6734 

4 

57 

537 

•308  186 

74£ 

•341  200 

553 

797C 

9277 

9393 

3 

58 

815 

468 

•325018C 

473 

824 

•374  0671  -390  195£ 

•406  2051 

2 

59 

•292093 

740 

2931 

746S 

•358  096J 

3369    463£ 

470S 

1 

60 

•309  017 

568$ 

•342020 

367 

60661   731] 

7366 

0 

/ 

73'1 

72° 

71° 

70° 

69° 

68°  j  67° 

66°  |  ' 

NAT.  COSINE. 


NATURAL   SINES. 


247 


/ 

24° 

25°        26°        27° 

28°        29°   |     30° 

31° 

/ 

0 

•406  7300 

•4226183 

•4383711 

•4:,:;  nw:> 

•469  4716  -484  8096  -500  0000 

515  03S1 

00 

1 

•407  0024 

8819 

6326 

•4542497 

7  2s  4 

•485  06401         2519 

2874 

59 

2 

2681 

•423  1455 

8940 

5  us  si        9S52 

3184         5037 

5307 

58 

2 

6337 

40.0 

•439  1553 

707'.'  -470  2419 

5727         7556 

7859 

57 

4 

7993 

6725 

4166 

•4550269 

4986 

8270-5010973 

5160:551 

56 

5 

•408  O04'.» 

9:jt50 

8779 

2859 

7;V,:;:-4SOOS12l         2591 

2842 

55 

6 

3305 

424  l'.H>4 

9392 

544'.)  -47  Id  11'.' 

:«.">4 

5107 

5333 

54 

7 

5980 

4628 

•44020.>t 

2685 

5895 

7024 

7824 

53 

8 

8615 

7262 

4615 

•456  0027 

5250 

S43'i;  -502  0140 

5170:314 

52 

9 

•409  1269 

9895 

7227 

3216 

7815 

487  0977  1         2655 

2S04 

51 

10 

302:; 

425  2.V2S 

9838 

5804 

472  0380 

3517         5170 

529.". 

50 

11  ;         6577 

5161 

•441  2  Us 

8392 

2944 

6057          7«X-, 

7782 

49 

12           92:  jo 
13   -4101SS:5 

7793 
•4260425 

50:>!» 
7668 

•457  0979 
356(1 

5508 
8071 

8597  -503  0199 
488  1136         2713 

518  0270 
2758 

48 
47 

14 

4536 

3056 

•442  027  S 

6153 

•473  0634 

3674         5227 

5246 

46 

16 

7189 

56S7 

2SS7 

8789 

3197 

6212         7740 

77:;.", 

45 

16 

9841 

831  S 

5496 

•458  1325 

5759 

8750-5040252 

5190219 

44 

17 

•4112192 

•4270949 

8104 

8910 

§321 

489  1288 

2765 

2705 

43 

IS 

5L44 

8579 

•4430712 

6496 

•47408S2 

3825 

5270 

•5191 

42 

1'J 

7795 

62'.  is 

8319 

9080 

3443 

6361 

77SS 

7070 

41 

2o 

•412  0445 

8S38 

5927 

•459  1665 

6004 

8897 

•505  0298 

520  0161 

40 

21 

S096 

•428  1407 

8534 

42  is 

8564 

490  1433 

2809 

2040 

39 

22 

5745 

4095 

•4441140 

6832 

•475  1124 

3968 

5:519 

5130 

38 

23 

8395 

6723 

8746 

9415 

8683 

6503 

7828 

7613 

37 

24 

•4131  OH 

9351         6352 

•460  199S 

0242 

9038 

•50603:5* 

521  00'.  Mi 

36 

25 

3693 

•429V.I7U,         8957 

4680 

8801 

491  1572 

2840 

2579 

35 

B8 

6342 

46061-445  1562 

7162 

•476  1359 

4105 

63ftf 

5001 

34 

27 

8990 

72:;:; 

416? 

9744 

3917 

8638 

7863 

7541 

33 

28 

•414  io:;s 

9859 

6771 

461  2325 

6474 

9171 

•507  0370 

5220024 

32 

23 

4285 

•4302485 

9375 

4906 

9031 

•492  1704 

2877 

250.- 

31 

88 

6932 

5111 

•446  1978 

74SC, 

•477  15SS 

4286 

5384 

4981 

30 

81 

9579 

7738 

4.")  si 

•462  0066 

4144 

6707 

789 

7401 

29 

32 

•415  2226 

•431  OSiil 

7184 

2646 

6700 

9298 

•508  0390 

994* 

28 

S3 

4S72 

2986 

978fl 

6228 

'.)_'.,: 

•4931829 

2901 

•r>2:>.2»2 

27 

31 

7517 

5610 

•447  2:iss 

7804 

•478  1810 

4359 

540< 

4'.Hi: 

26 

•"<> 

•416  0103 

82:;  4 

4991 

•463  0382 

4864 

6889 

79U 

73S1 

25 

86 

2808 

•432  0857 

7591 

'2960 

691! 

9419 

•5090414 

9S5! 

24 

SI 

5453 

3481 

•448  0192 

6538 

9472 

•494  194H 

2.1  is 

W42:;:;< 

•X 

S8 

8097 

6103 

2792 

8115 

•479  202i 

447' 

5421 

481: 

22 

80 

•417  0741 

87  2'  i 

5892 

464  0692 

457'. 

7005 

7924 

729! 

21 

40 

8385 

•433  134S 

7992 

3269 

7131 

9532J-5100426 

97« 

20 

41 

'      6023 

3970 

•449  <>f.:il 

6845 

90s: 

•4952060 

292s 

•525224' 

19 

42 

8671 

6591 

319! 

8420 

•480  2235 

4587 

542: 

4717 

IS 

4:; 

•41S  i:!i:! 

9212 

5789 

•465  0990 

478b 

7113 

793i 

7191 

17 

41 

3956 

•434  1832 

83S7 

3571 

7:i:  '.7 

9639 

5110431 

9661 

16 

4:. 

6597 

4463 

•4500-.IS4 

6145 

9888 

•496  2105 

2.i:il 

•52021:;: 

15 

46 

923  ) 

7072 

35S2 

8719 

•481  2438 

40',R) 

5431 

401: 

14 

47 

•419  1S.SO 

6179 

•4661293 

4987 

7215 

799 

7osr 

13 

4S 

4521 

•4352311 

8775 

3806 

7537 

9740 

•512  042! 

955^ 

12 

4'.» 

7101 

4:»::  > 

•451  1372 

6439 

•482  0086 

•497  2204 

2927 

•527  2031 

11 

M 

9301 

7548 

3967 

9012 

2684 

4787 

542f 

450: 

10 

61 

•420  2441 

•436  0106 

6563 

•467  1584 

5182 

7310 

792 

697: 

9 

52 

50SO 

2784 

9158 

4156 

779 

9833 

•5130420 

941: 

8 

53 

77T.I 

5401 

•452  1  75." 

6727 

•483  0277  -49S  2:455 

2916 

•528  1914 

7 

54 

•421  o:;:,s 

8018 

4347 

9298 

2S24I         4877 

541: 

438; 

6 

55 

2996 

•437  0034 

6941 

•4681809 

5370          7399 

7908 

C.S.V 

5 

56 

5634 

8251 

953J 

44:;:* 

7916         9320 

•514  0404 

9322 

4 

57 

8272 

5866 

•453  2128 

7009 

•4840462-4992441 

2894 

•529  179( 

3 

58 

•422ir.ii):) 

84S2 

4721 

9578 

3007  i         4961 

639! 

425* 

2 

59 

3546 

•4381097 

7313 

•4692147 

r.5521         7481 

7887 

072' 

1 

BO 

61  S3 

3711 

9905 

47  10 

8096  -500  0000 

•5150381 

919: 

0 

/ 

65° 

6i° 

63° 

02° 

61° 

60° 

59° 

58° 

/ 

NAT.  COSINE. 


246 


NATURAL   SINES. 


10° 

\1°   \    18° 

19° 

2C° 

21°    1    22° 

28°       ' 

.275  C374 

292  3717 

309  0170 

325  5082 

342  0201 

358  3679 

3746066-3907311    60 

9170 

C499 

2936 

8432 

2935 

6395 

8763         9989    59 

•276  19U5 

92SO 

5702 

326  1182 

5CC8 

9110 

3751459.3912066   58 

4701 

293  2061 

84C8 

3932 

8400 

359  1125 

4156         5313   57 

755i 

4842 

3101234 

6681 

343  1133 

4540 

6852         8019   56 

"277  0352 

7023 

3999 

9430 

3805 

7254 

9547  -392  0095'  55 

3147 

294  0403 

67  C4 

327  2179 

6597 

9968  -376  2243         3371   54 

5941 

3183 

9529 

4928 

9329 

36021821        4938         C047    53 

8731 

5963 

311  221)4 

7670 

3442000 

5395;        7032         8722;  52 

•278  1530 

8743 

5058 

m0424 

4791 

810S|-377  0327 

•3931397'  51 

4324 

295  1522 

„  .7822 

3172 

7521 

361  0£21 

3021 

4071    50 

7118 

4302 

3120586 

5919 

345  0252 

3534 

5714 

6745   49 

9911 

7081 

3349 

8666 

2982 

6246 

8408 

9419 

48 

•279  2704 

9859 

6112 

329  1413 

5712 

8958 

378  110L-394  2093 

47 

5497 

2962638 

8875 

4160 

8441 

362  1069 

37941         4766 

46 

8290 

5416 

313  1G3N 

6906 

3461171 

4380 

64S6|        7439 

45 

•280  10S3 

8194 

4400 

9653 

3900 

7091 

9178-3950111 

44 

3875 

297  0971 

71C3 

330  2398 

6628 

9802 

379  1870         2783 

43 

6667 

3749 

9925 

5144 

9357 

363  2512 

4562         5455 

42 

9459 

6526 

3142681 

7889 

347  2085 

5222 

7253         8127 

41 

•281  2251 

9303 

544-c 

331  0634 

4812 

7932 

9944  -396  0798 

40 

5042 

298  2079 

8209 

3379 

7540 

3640641 

380  2034 

3468 

39 

7833 

4856 

315  0909 

6123 

348  0207 

3351 

5324 

6139 

38 

•282  0624 

7632 

3730 

8867 

2994 

6059 

8014 

8809 

37 

3415 

2990408 

6490 

3321011 

5720 

8768 

381  0704 

•397  1479 

36 

6205 

3184 

9250 

4355 

8447 

365  1476 

3393 

4148 

35 

8995 

5959 

316  2010 

7098 

349  1173 

4184 

6082 

6818 

34 

•283178;; 

8734 

4770 

9841 

3898 

CS91 

8770 

9486 

33 

4575 

300  1509 

7529 

333  2584 

6624 

9599 

382  1459 

•398  2155 

32 

7364 

4284 

317  0288 

6326 

9349 

366  2306 

4147 

4823 

31 

•2840153 

7058 

3047 

8069 

350  2074 

5012 

6834 

7491 

30 

2942 

9832 

5805 

3340810 

4798 

7719 

9522 

.3990158 

29 

5731 

301  2COO 

8563 

3552 

7523 

367  0425 

•383  2209 

2825 

28 

8520 

5380 

318  1321 

6293 

351  0240 

3130 

4895 

5492 

27 

•285  1308 

8153 

4079 

9034 

2970 

5836 

7582 

8158 

26 

4096 

•3020926 

683C 

335  1775 

5693 

8541 

•3840268 

•400  0825 

25 

6884 

3699 

9593 

4511 

8416 

368  1246 

2953 

349(1 

24 

9671 

6471 

319  2350 

725C 

3521139 

3950 

5039 

ofce 

23 

•2862458 

9244 

51  OC 

9996 

3862 

6654 

8324 

8821 

22 

524C 

•303  201G 

7863 

336  2735 

6584 

9358 

•385  1008 

•401  148t 

21 

8032 

4788 

320  0619 

5475 

9300 

•369  2061 

3093 

4150 

20 

•287  0819 

-  7559 

3374 

8214 

3532027 

4765 

6377 

C814 

19 

3605 

•3040331 

6130 

•337  0953 

4748 

7468 

906C 

9478 

18 

6391 

3102 

8885 

3691 

7469 

•370  0170 

•386  1744 

•402  2141 

17 

9177 

5872 

•321  1C4C 

6429 

3540190 

2872 

4427 

4804 

16 

•288  1963 

8G43 

439f 

9167 

2911 

5574 

7110 

7467 

15 

4748 

•305  1413 

7141 

•338  1905 

5630 

8276 

9792 

•4030129 

14 

7533 

4183 

9903 

4642 

8350 

•371  0977 

•387  2474 

2791 

13 

•289  0318 

6953 

•3222657 

7379 

•355  107t 

307  S 

5150 

5453 

12 

3103 

9723 

5411 

•3390110 

3781 

6379 

7837 

8114 

11 

5887 

•306  2491. 

8164 

2852 

6508 

9079 

•3880518 

•404  0775 

10 

8671 

5201 

•3230917 

5589 

922f 

•3721780 

3199 

343C 

9 

•2901450 

8030 

3670 

8325 

•3561944 

4479 

5880 

€096 

8 

4239 

•307  0798 

642L 

•340  1061 

4C62 

7179 

8560 

875C 

7 

702L 

3566 

9174 

3796 

738( 

9878 

•389  1240 

•405  1416 

6 

980^ 

6334 

•3241926 

6531 

•357  0097 

•373  2577 

3919 

4075 

5 

•291  258F 

910'. 

4678 

9265 

2814 

5275 

659S 

6734 

4 

537 

•308  1869 

7429 

•341  2000 

5531 

7973 

9277 

9393 

3 

815S 

463C 

•325  01SC 

4734 

8248 

•374  0671  ,-3901955 

•406  2051 

2 

•292  093 

7403 

2931 

7468 

•358  0964 

3369         403S 

470Q 

1 

37r 

•309  017U 

5682 

•3420201 

3679 

60661        7311 

736C 

0 

7o° 

72° 

71° 

70° 

69° 

68°    i    67° 

66° 

/ 

NAT.  COSINE. 


NATURAL   SINES. 


247 


24° 

25°    i    26° 

27° 

28°        29°        30° 

81° 

/ 

•406  7366 
•407  0024 

•422  6183 

8819 

•4383711 
6326 

4531)905  -469  4716  -484  8096'  -500  0000 
4542497:         7284-48506401         2519 

515  0381 
2874 

ro 

59 

2681 

•423  1455 

8940 

50881        9852 

3184J         5037 

53<'.7 

58 

5337 

4090 

•439  1553 

7079  -470  2419 

57271         7556 

7859 

57 

7993 

6725 

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455  0263 

49S»i 

8270-5010073 

5160351 

50 

•408  0049 

9360 

6779 

2859 

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2842 

f.5 

3305 

424  1994 

9392 

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5333 

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4628 

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8038 

2685 

58951         7024 

7824 

53 

8615 

7282 

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456  0627 

5250 

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5020140 

5170314 

52 

•409  1269 

9895 

7227 

3216 

7815 

487  0977 

2655 

2804 

61 

3920 

425  2528 

9838 

5804 

472  0380 

3517 

5170 

529.". 

50 

6577 

5161 

•441  2  44S 

8392 

2944 

6057 

7685 

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9230 

7793 

5059. 

457  0979 

5508 

8597 

503  0199 

518  0270 

48 

•4101883 

•4260425 

7668 

3560 

8071 

488  1136 

2713 

2758 

47 

4536 

3050 

•4420278 

6153 

473  0634 

3674 

5227 

5240 

46 

71S9 

56S7 

2887 

8739 

3197 

6212 

7740 

7733 

45 

9841 

8318 

54'.K1 

458  1325 

5759 

8750 

•504  0252 

519  0219 

44 

•411  2492 

•4270949 

8104 

3910 

8.321 

489  1288 

2765 

2705 

43 

5144 

3579 

•4430712 

6496 

474  0882 

3825 

5276 

•5191 

42 

7795 

6208 

3319 

9080 

3443 

6361 

7788 

7676 

41 

•412  0445 

8838 

5927 

459  1665 

6004 

8897 

•505  0298 

520  0161 

40 

3096 

•428  1467 

8534 

4248 

8564 

490  1433 

2809 

2(34(1 

39 

5745 

4095 

•4441140 

6832 

•475  1124 

3968 

5319 

5130 

38 

8395 

6723 

3740 

9415 

3683 

6503 

7828 

7613 

37 

•4131041 

9351 

6352 

460  1998 

6242 

9038 

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521  0096 

30 

3693 

•429  1979 

8957 

4580 

8801 

491  1572 

2846 

2579 

35 

6342 

4liOi)l  -445  1562 

7162 

•476  1359 

4105 

5355 

5061 

34 

8990 

7233 

4107 

9744 

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7  863 

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33 

•414  1638 

9859 

6771 

461  2325 

6474 

9171 

•507  0370 

•522  0024 

32 

42S5 

•4302485 

9375 

4906 

9031 

•492  1704 

2877 

2505 

31 

6932 

5111 

•446  1978 

7486 

•477  1588 

4236 

5384 

4986 

30 

9579 

773ii 

4581 

•462  0066 

4144 

6767 

7890 

7461 

29 

•415  2226 

•431  0361 

7184 

2646 

6700 

9298 

•508  0390 

994f 

28 

4872 

2386 

9786 

5225 

9255 

•493  1829 

2901 

•523  2424 

27 

7517 

5610 

•447  2388 

7804 

•478  1810 

4359 

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26 

•416  0163 

8234 

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•463  0382 

4364 

6889 

7910 

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25 

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5538 

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2918 

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23 

8037 

6103 

2792 

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4476 

5421 

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•417  0741 

8720 

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4579 

7005 

7924 

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21 

3385 

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7992 

3269 

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97  6f 

20 

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5845 

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292s 

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19 

8671 

6591 

3190 

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4587 

5421 

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9212 

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4786 

7113 

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8387 

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16 

6597 

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•450  0984 

6145 

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•496  2165 

2931 

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15 

9231 

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9692 

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8775 

3866 

7537 

9740 

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9558 

12 

7161 

4930 

•451  1372 

6439 

•482  0086 

•497  2264 

2927 

•527  2030 

11 

9301 

7548 

3967 

9012 

2634 

4787 

5425 

450? 

10 

•420  2441 

•436  0166 

6563 

•467  1584 

5182 

7310 

7923 

697: 

9 

5080 

2784 

9158 

4156 

7730 

9833 

•513  0420 

944! 

8 

7719 

5401 

•4521753 

6727 

•483  0277 

•498  2355 

2916 

•528  1914 

7 

•421  0358 

8018 

4347 

9298 

2824 

4877 

5413 

438: 

6 

2996 

•437  0031 

6941 

•4681869 

5370 

7399 

7908 

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5 

5634 

325  1 

9535 

4439 

7916 

9920 

•514  0404 

932i 

4 

8272 

5800 

•453  2128 

7009 

•4840462-4992441 

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•529  1791 

3 

•4220909 

8482 

4721 

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3007 

4961 

5393 

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2 

3546 

•4381097 

7313 

•469  2147 

6552 

7481 

7887 

67  2f 

1 

6183 

3711 

9905 

4716 

8096  -500  0000 

•515  0381 

9193 

0 

65°     |     64° 

G3° 

62° 

61° 

00° 

59° 

58° 

/ 

NAT.  COSINE. 


248 


NATURAL   SINES. 


/ 

32°   |    33° 

34° 

35° 

36° 

37° 

38° 

39° 

t 

0 

529  9193-544  6390 

5591929 

573  5764 

587  7853 

•601  8150 

615  6615  -629  3204 

00 

1 

2 
3 

5301659         8830 
4125-545  1269 
6591         3707 

4340 
6751 

916-2 

8147 
574  0529 
2911 

5880206 
2558 
4910 

602.0473 
2795 
5117 

8907i         5464 
6161198!         7724 
3489         9983 

59 

58 

57 

4 

9057 

'    6145 

560  1572 

5292 

7262 

7439 

5780/630  2242 

56 

5 

531  1521 

8583 

3981 

7672 

9613 

97  CO 

80691        4500 

56 

6 

3986 

546  1020 

6390 

575  0053 

589  1964 

•603  2080 

617  0359 

6758 

54 

7 

6450 

3456 

879S 

2432 

4314 

4400 

2648 

9015 

53 

8 

8913 

5892 

561  1206 

4811 

6663 

6719 

4936 

631  1272 

52 

9 

5321370 

8328 

3614 

7190 

9012 

9038 

7224 

3528 

51 

10 

3839 

547  0763 

•     6021 

9568 

•590  1361 

.604  1356 

9511 

5784 

50 

11 

6301 

3198 

8428 

576.1946 

3709 

3674 

618  1798 

8039 

49 

12 

8763 

5632 

5620834 

4323 

6057 

5991 

4084 

632  0293 

4S 

18 

5331224 

8066 

3239 

6700 

8404 

8308 

6370 

2547 

47 

14 

3685 

548  0499 

5645 

9076 

•591  0750 

•605  0624 

8655 

4800 

46 

15 

6145 

2932 

8049 

•577  1452 

3096 

2940 

619  0939 

7053 

45 

16 

8605 

5365 

563  0453 

3827 

5442 

5255 

3224 

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44 

17 

5341065 

7797 

2857 

6202 

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633  1557 

43 

IS 

352.3 

5490228 

5260 

8576 

•5920132 

9884 

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19 

5982 

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•578  0950 

2476 

•606  2198 

620  0073 

6059 

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20 

8440 

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5640066 

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40 

21 

5350898 

7520 

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39 

2-2 

3355 

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4869 

8069 

9505 

9136 

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00 

23 

5812 

550  2379 

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•593  1847 

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9198 

5057 

37 

24 

8268 

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7305 

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25 

5360724 

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565  2070 

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551  2091 

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8314 

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3550 

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6292 

32 

29 

•537  0543 

6944 

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4661 

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31 

30 

2996 

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7030 

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2904 

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9698 

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6645 

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•5390158 

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40 

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•542  1971 

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4294|         4893 

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5566 

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8 

53 

9302 

5036 

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5 

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0943 

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1 

60 

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7853         8150 

6615 

•629  3204 

7876 

0 

/ 

57° 

56° 

55° 

54° 

53° 

52°. 

51° 

50° 

/ 

NAT.   COSINE. 


NATURAL   SINES. 


249 


40° 

41° 

42° 

43°   44° 

45° 

46° 

47° 

/ 

•6427876 

6560590 

669  1306 

681  9984  -694  6584 

707  1068 

193398 

731  3537 

60 

•b430104 

2785 

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68221111    8676 

3124 

6418 

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2332 

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5180 

7438 

7503 

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7174 

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6363 

285'8 

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57 

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9367 

9948 

8489 

4949 

9291 

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"32  1467 

56 

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657  15CO 

670  2108 

6830613 

7039 

708  1345 

3494 

3449 

55 

•6441236 

3752 

4266 

2738 

9128 

3398 

5511 

5429 

54 

3461 

5944 

6424 

4861 

69C1217 

5451 

7528 

7409 

53 

5685 

8135 

8682 

6984 

2306 

7504 

9544 

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7909 

6580326 

671  0739 

9107 

5392 

9556 

21  1559 

733  1367 

51 

•645  0132 

2516 

2895 

684  1229 

'7479 

7091  €07 

3574 

3345 

50 

2355 

4706 

5051 

3350 

9565J    3657 

5589 

5322 

49 

4577 

6895 

7206 

5471 

697  1651    5707 

7C02 

7299 

48 

6798 

9083 

9361 

7591 

3736    7757 

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47 

9019 

0591271 

6721515 

9711 

58211    9806 

-221C28 

734  1250 

46 

•6461240 

3458 

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685  1830 

79051-7101854 

3640 

3225 

45 

3460 

5G45 

5821 

3948 

9988J    3901 

5651 

5199 

44 

5679 

7831 

7973 

6066 

698  2071    6948 

7C61 

7173 

43 

7898 

6600017 

6730125 

8184 

4153    7995 

9671 

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•647  0116 

2202 

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62341-711  0041 

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41 

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6570 

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699  0396    4130 

5698 

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661  0936 

674  0876 

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6628 

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9662 

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11 

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316 

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618 

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9 

301 

399 

294 

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49° 

48° 

47° 

46° 

45° 

44° 

43°   42° 

NAT.  COSINE 


NATURAL   SINES. 


48°    49° 

50°    51°    52° 

53° 

54° 

•713  1448  :  -754  7090 

•766  0444  -777  1460  -788  0108 

798  6355 

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3394     9004 

2314     3290     1898 

8105 

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5340  -7550911 

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7285  :    2818 

6051  ;    6949 

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9229  i    4724 

7918     8777 

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3517     4258 

2627 

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800  0338 

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4240 

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2821 

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•70S  0973  -7791557 

9707 

5571 

8930 

4760 

9951 

2835 

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•790  1550 

7314 

811  0038 

6099 

•757  1851 

4697 

5202 

3333 

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6558     7024 

5115 

801  0797 

4040 

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5650 

8418  |    8845 

6896 

2538 

5740 

2510 

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•769  0278  ;  -780  0005 

8676 

4278 

7439 

4446 

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2137  1    2485 

•791  0456 

6018 

9137 

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•758  1343 

3990 

4304 

2235 

7756 

•812083*5 

8317 

3240 

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6123 

4014 

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•747  0251 

5136 

7710 

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802  1232 

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7031 

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2969 

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•770  1423  ;  '781  1574 

9345 

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•759  0820 

3278     3390 

•7921121 

6440 

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5132     5205 

289(1 

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•813  1008 

9912 

4000 

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•748  1842 

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8840     8833 

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•803  1042 

4393 

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•771  0692  :  '782  0046 

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•8040299 

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2028 

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•7721794 

•783  1511 

8843 

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3042 

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5484 

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9187 

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2379 

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•7501111 

5383 

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•815  1278 

3034 

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9182 

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•805  0004 

2903 

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9152 

•7731027  j  '784  0547 

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•762  1036 

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•8161370 

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3050 

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•785  1368 

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2720 

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4446 

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4204 

5767 

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4149 

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•817  1449 

6005 

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•775  1283 

•7860367 

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•807  1321 

3125 

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•7641714 

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2165 

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3038 

4801 

9894 

3590 

4957 

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•797  0572 

4754 

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•7531808 

5465 

6794 

5759 

2:  ;•_*.> 

6470 

8151 

3721 

7340 

8029 

7555 

4084 

8185 

9824 

5034 

9214 

•776  0464 

9350 

5839 

9899 

•818  1497 

7546 

•765  1087 

2298 

•787  1145 

7594 

•808  1012 

3109 

9457 

2900 

4132 

2939 

9347 

3325 

4841 

•754-1368 

4832 

5905 

4732 

•7981100 

5037 

0512 

3278 

6704 

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2853 

6749 

8182 

5187 

8574 

9629 

8316 

4004 

8400 

9852 

7096 

•766  0444 

•777  1460 

•788  0108 

0355 

•8090170 

•8191520 

41° 

40° 

39° 

38° 

37° 

36° 

35° 

NAT.   COSINE. 


NATURAL   SINES. 


251 


55° 

56° 

57° 

58° 

59° 

60° 

61° 

/ 

•819  1520 

829  0376 

838  6706 

848  0481 

857  1673 

8660254 

874  6197 

60 

3189 

2002 

8290 

2022 

3171 

1708 

7607 

59 

4850 

3628 

9873 

3562 

4668 

3161 

9016 

58 

6523 

5252 

839  1455 

5102 

6164 

4614 

875  0425 

57 

8189 

6877 

3037 

6641 

76CO 

C006 

1832 

56 

9854 

8500 

4618 

8179 

9155 

7517 

3239 

55 

•820  1519 

830  0123 

6199 

9717 

858  0649 

8967 

4645 

54 

3183 

1745 

7778 

849  1254 

2143 

867  0417 

6051 

53 

484(3 

336(3 

9357 

2790 

3635 

1866 

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52 

6509 

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840  0936 

4325 

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.  3314 

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51 

8170 

6607 

2513 

5860 

6619 

4762 

8760263 

50 

983:2 

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4090 

7394 

8109 

6209 

1665 

49 

•821  1492 

9845 

5066 

8927 

9599 

7655 

3067 

•18 

3152 

831  1463 

7241 

850  0459 

859  1088 

9100 

4468 

47 

4811 

3080 

8816 

1991 

2576 

868  0544 

5868 

46 

6469 

4696 

841  0390 

3522 

4064 

1988 

7268 

45 

8127 

6312 

1963 

5053 

5551 

3431 

8666 

44 

9784 

7927 

8536 

6582 

7037 

4874 

877  0064 

43 

•822  1440 

9541 

5108 

8111 

8523 

6315 

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42 

3096 

8321155 

6679 

9639 

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!    4751 

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9196 

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40 

6405 

4380 

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2693 

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4219 

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38 

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9212 

4524 

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833  0822 

6091 

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35 

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•861  0380 

'.821 

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84 

6316 

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1839 

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9256 

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33 

7965 

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8430787 

3360 

3337 

•8700691 

5394 

32 

9614 

7252 

2351 

4881 

4815 

2124 

6783 

31 

•824  12(32 

8858 

3914 

6402 

6292 

3557 

8171 

30 

2909 

834  0463 

5477 

7921 

7768 

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29 

4556 

2068 

7039 

9440 

9243 

6420 

•879  0946 

28 

6202 

3672 

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7851 

2332 

27 

7847 

5275 

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2475 

2191 

9281 

3717 

26 

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1720 

3992 

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5102 

25 

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8479 

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5137 

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24 

2778 

•835  0080 

4838 

7023 

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23 

4420 

1680 

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8538 

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9251 

22 

6062 

3279 

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•8540051 

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21 

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20 

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•826  0983 

8074 

2618 

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18 

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9670 

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6152 

17 

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•8361266 

5726 

7609 

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3538 

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16 

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2862 

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15 

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14 

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7801 

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13 

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7643 

1932 

3643 

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12 

2440 

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5149 

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11 

4074 

•837  0827 

5030 

6655 

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2058 

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10 

5708 

2418 

6579 

8160 

7134 

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7155 

9 

7340 

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4891 

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8 

8971 

5598 

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6307 

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7 

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7187 

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2IJ71 

1514 

7722 

•8821269 

6 

2234 

8775 

2765 

4173 

2973 

9137 

2638 

5 

3864 

•838  0363 

4309 

5674 

443f 

•8740550 

4007 

4 

5493 

1950 

5853 

7175 

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1963 

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3 

7121 

3536 

7397 

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7344 

3375 

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2 

8749 

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•857  0174 

8799 

4786 

8110 

1 

•829  0376 

6706 

•848  0481 

1673 

•8660254 

6197 

9476 

0 

34° 

33° 

82° 

31° 

30° 

29' 

28° 

/ 

NAT.   COSINE. 


252 


NATURAL   SINES. 


62° 

63° 

04° 

65° 

06° 

67°. 

68° 

•882  9476 

.891  OOC5  (  -898  7940 

•9063078 

•913  5455 

•920  5049 

•927  1839 

•8830841 

13S5 

9215 

4307 

6C37 

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2928 

2206 

2705 

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5535 

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7320 

4016 

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7277 

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2540 

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8363 

9017 

9291 

6848 

1665 

3718 

2986 

9447 

•884  0377 

•8920C06 

8117 

2888 

4895 

4116 

•928  0531 

1736 

1920 

9386 

4111 

6072 

6246 

1014 

3095 

3234 

•900  0654 

5333 

7247 

6375 

2696 

4453 

4546 

1921 

6554 

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5810 

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5938 

8522 

8480 

5718 

•9080214 

1943 

•9220884 

7017 

9876 

9789 

6982 

1432 

3115 

2010 

8096 

•885  1230 

•8931098 

8246 

2C49 

4286 

3134 

9173 

2584 

2406 

9508 

3866 

6456 

4258 

•9290250 

3936 

3714 

•901  0770 

6082 

6626 

6381 

1326 

5288 

6021 

2031 

6297 

7795 

6503 

2401 

6639 

6326 

3292 

7511 

89C3 

7624 

3475 

7989 

7632 

4551 

8725 

•9160130 

8745 

4549 

9339 

8936 

5810 

9938 

12D7 

9865 

5C22 

•886  0688 

•8940240 

7068 

•909  1150 

24C2 

•92309S4 

6094 

2036 

1542 

8325 

2561 

3627 

2102 

7705 

3383 

2844 

9582 

3572 

4791 

3220 

8835 

4730 

4146 

•9020838 

4781 

5955 

4330 

9905 

6075 

5446 

2092 

5990 

7118 

5452 

•9300974 

7420 

6746 

3347 

7199 

8279 

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2042 

8765 

8045 

4600 

8406 

9440 

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3109 

•887  0108 

9344 

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9613 

•917  OC01 

8795 

4176 

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•895  0041 

7105 

•9100819 

17  CO 

9908 

5241 

2793 

1938 

8356 

2024 

2919 

•924  1020 

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4134 

3234 

9606 

3228 

4077 

2131 

7370 

5475 

4529 

•903  0856 

4432 

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3242 

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6035 

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4351 

9496 

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3353 

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•8890171 

8727 

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7620 

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1133 

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•9190207 

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2186 

4164 

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8271 

1201 

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3238 

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9509 

2393 

2499 

9805 

4290 

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•905  0746 

3584 

3644 

•926  0902 

5340 

8149 

6433 

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4775 

4788 

2000 

6390 

9476 

7715 

3219 

5965 

5931 

3096 

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•890  0803 

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4454 

7154 

7073 

4192 

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2128 

•898  0276 

5688 

8342 

8215 

5286 

9535 

3453 

1555 

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•933  0582 

4777 

2834 

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•9130716 

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6100 

4112 

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1902 

1035 

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•9060618 

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2774 

9658 

3718 

8744 

6665 

1848 

4271 

3912 

•927  0748 

4701 

•891  0065 

7940 

3078 

5455 

5049 

1839 

5804 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

NAT.  COSINE. 


NATURAL  SINES. 


253 


69° 

70°    71° 

72° 

73° 

74° 

75° 

•933  5804 
6846 

•939  6926 
7921 

•945  5186 
6132 

951  0565 
1464 

9563048 
3898 

961  2617 
3418 

965  9258 
9660011 

7888 

8914  1    7078 

2361 

4747 

4219 

0762 

8928 

9907 

8023 

3258 

5595 

5019 

1513 

9968 

•940  0899 

89(38 

4154 

6443 

5818 

2263 

•934  1007 

1891 

9311 

5050 

7290 

6616 

3012 

2045 

288> 

•9460854 

5944 

8136 

7413 

3761 

3082 

3871 

1795 

6838 

8981 

8210 

4508 

4119 

4860 

2736 

7731 

9825 

9005 

5255 

5154 

5848 

3677 

8623 

957  0669 

9800 

6001 

6189 

6835 

4616 

9514 

1512 

9620594 

6746 

7223 

7822 

5555 

952  0404 

2354 

1387 

7490 

8257 

8808 

6493 

1294 

3195 

2180 

8234 

9289 

9793 

7430 

2183 

4035 

2972 

8977 

•9350321 

•941  0777 

8366 

3071 

4875 

3762 

9718 

1352 

1760 

9301 

3958 

5714 

4552 

967  0459 

2382 

2743  1  -947  0236 

4844 

6552 

5342 

1200 

3412 

3724 

1170 

5730 

7389 

6130 

1939 

4440 

4705 

2103 

6615 

8225 

6917 

2678 

5468 

5686 

3035 

7499 

9060 

7704 

3415 

6495 

6665 

3966 

8382 

9895 

8490 

4152 

7521 

7644 

4897 

9264 

958  0729 

9275 

4888 

8547 

8621    5827 

9530146 

1562 

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5624 

9571 

9598  !    6756 

1027 

2394 

0843 

6358 

•936  0595 

•942  0575 

7684 

1907 

3226 

1626 

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1618 

1550 

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4056 

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2525     9538 

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101 

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219 

181 

850 

2253 

6920 

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•951  0565 

304 

261 

925 

2957 

20° 

19° 

18° 

17° 

16° 

15° 

14° 

NAT,  COSINE. 


254 


NATURAL  SINES. 


/ 

76° 

77° 

78°  1  79° 

80° 

81° 

82° 

/ 

0 

•970  2957 

•974  3701 

•978  1476 

•981  6272 

•9848  078 

•9876  883 

•9902  681 

60 

1 

36  0 

4355 

2080 

6826 

582  -9877  338 

•9903  085 

59 

2 

4363 

5008 

2684 

7380 

•9849  086  1     792 

489 

58 

3 

5065 

5660 

3287 

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589  j  '9878  245 

'  891 

57 

4 

5766 

6311 

'   3889 

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•9850091     GJI7 

9904  293 

56 

5 

6466 

6962 

4490 

9037 

593  1  -9879  148 

694  '  55 

6 

7165 

7612 

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9587  -9851093     599 

9905  095  54 

7 

7863 

8261 

5689 

•9820137      593  -9880048 

494 

53 

8 

8561 

8909 

6288 

0686 

•9852092 

497 

893 

52 

9 

9258 

9556 

6886 

1234 

590 

945 

9906  290 

51 

10 

9953 

•975  0203 

7483 

1781 

•9853  087 

•9881  392 

687 

50 

11 

•971  0649 

0849 

8079 

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4369 

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6502 

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5739/507  3290 

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2 

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3726-509  1591  -531  3304 

9288 

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7326         5254 

7094  -554  3091 

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8606 

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65° 

64°          63° 

62° 

61° 

60°     |     59° 

68°     |    ' 

KAT.  COTAN. 


258                                    NATURAL    TANGENTS. 

/  i 

10° 

17° 

18° 

19° 

20° 

O|o 

22°    1    23° 

/ 

0 

2867454 

305  7307 

324  9197 

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383  8640 

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60 

1 

287  0602 

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325  241;, 

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364  299', 

384  1978 

36461         8182 

59 

2 

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405  0417 

5051 

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4 

288  0050 

307  0034 

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6296 

365  2886 

385  1996 

3804 

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66 

5 

3201 

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426  1924 

55 

6 

6352 

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3271724 

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386  2021 

3968 

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8 

289  2655 

308  2771 

4941 

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347  2586 

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323  1387 

5840 

367  2680 

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73° 

72° 

71° 

70° 

69° 

68° 

67° 

66° 

/ 

NAT.  COTAN. 


NATURAL   TANGENTS. 


259 


24° 

25° 

26° 

27°    |    28° 

29° 

30° 

31° 

r 

•445  2287 

•466  3077 

•487  7326 

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6008606 

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5773 

6618  4880927 

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601  2566 

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4530 

510  2585 

4559 

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3705 

8133 

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7250  -489  1737 

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8311 

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5765 

5562119-5792912 

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55 

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4342         8949 

7259 

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5929 

6797 

603  2386 

54 

6708 

7890  -490  2557 

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9739 

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6354 

53 

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6166 

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557  3551 

4573 

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52 

3693 

4988 

9775 

8275 

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7364 

8462 

4294 

51 

7187 

8539  -491  33861-513  1950 

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5581179-5812353 

8266 

50 

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5025 

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6245 

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8811 

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7675 

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559  2629 

4034 

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47 

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6658          9440 

6449 

7930 

4170 

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4019         6943 

560  0269 
4091 

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8149 
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45 
44 

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7702-5380094 

7914 

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561  1738 

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8067 

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5683         7659-495  3171  '517  2441 
9188  -474  1222         6794|         6129 
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562  3219 
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566  1568J        4369 

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27 

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567  3098 

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8032 

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8357  :>479  1197 

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568  079l|        3984 

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5397 

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8918  '480  1932)        8547         8990 

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569  23391         5768 

4077    21 

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5512  -502  2189'-524  2698 

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20 

5962 

9093 

5832 

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570  0045 

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19 

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18 

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3829 

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17 

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6188 

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8188 

15 

3591 

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14 

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13 

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4189 

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7547 

7054I-5961196-6200263 

12 

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5140         4291 

11 

7710  '484  1368 

8668 

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10 

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4959 

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8 

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9633 

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7890 

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5996  -487  0126 

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7929 

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9536 
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3726 
7326 

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4648         4650     1 
8606          8694      o 

65° 

64° 

63° 

62° 

61° 

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59°          58° 

' 

KAT.  COTAN. 


NATURAL    TANGENTS. 


32° 

33°    1    34° 

35° 

36° 

37°       38° 

39°    |  ' 

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6494076-6745085 

"00  207;' 

726  5425 

53  5541  -781  2856 

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8212         9318 

6411 

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54  0102J        7542 

810  2658 

59 

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650  2350  -675  3553 

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727  431* 

4ur,r,  -782  2229 

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811  2300 

57 

4884 

651  0631  -676  2028 

9430 

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755  3799]  -783  1611 

7134 

56 

8935 

4774         6268 

"02  3773 

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8369 

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812  1951 

55 

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891S.-677  0509 

811!- 

729  2125 

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6780 

54 

7042 

6523064         4752 

703  2404 

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7514 

5700 

813  1611 

53 

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7211i         8997 

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730  1041 

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785  0400 

6444 

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5155 

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704  11CC 

5501 

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814  1280 

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758  1248 

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5829 

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705  4224 

8894 

759  0413 

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680  0246 

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12 

49 

23764 

91308 

89979 

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1-79  05121 

62955 

11 

50 

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1-5301023 

1-5900238 

33663 

1-7204736 

17362 

76003 

10 

51 

42210 

1074 

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44529 

16261 

29616 

89065 

9 

52 

51445 

20479 

207S3 

55405 

27797 

41SS3  1-8702141 

8 

53 

60638 

30219 

31070 

66292 

39346 

54102 

15231 

7 

54 

C9938 

39969 

41366 

77189 

50905 

60454 

28336 

6 

55 

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62477 

78759 

41455 

5 

56 

88463 

5949 

61987 

99016 

74060 

91077 

54588 

4 

57 

97738 

6927 

72512 

1-660994, 

85654 

1-80  03408 

67736 

3 

58  1-4307021 

7905 

82H47 

2088 

97260 

15751 

80898 

2 

59 

16311 

88S4 

92991 

31S3-J 

1-7308878 

2^103 

94074 

1 

60 

25610 

9865 

1-60  0334£ 

4279 

20508 

40473 

1-8307265 

0 

'  j  34° 

33° 

32° 

31°    30° 

29° 

28° 

/ 

NAT.  COTAN 


NATURAL   TANGENTS. 


62° 

63° 

64° 

65° 

66° 

67°  I  68° 

/ 

1-88  07265 

96  26105 

•05  03038 

2-1445069 

•24  60368 

•35  58524  2-47  50869 

60 

20470 

40227 

18185 

61366 

77962 

77590    71612 

59 

33690 

54364 

33349 

77683 

95580 

96683    92386 

58 

40924 

68518 

48531 

94021 

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•36  15801 

•48  13190 

57 

60172 

82688 

63732 

2-1510378 

30885 

34946 

b4023 

56 

73436 

96874 

78950 

26757 

48572 

54118 

54887' 

55 

86713 

9711077 

94187 

43156 

662*3 

73316 

75781 

54 

1-89  00006 

25296 

0609442 

59575 

81016 

92540 

96706 

53 

13313 

39531 

24716 

76015 

•2601773 

•37  11791 

•49  17660  |  52 

26635 

53782 

40008 

92476 

19554 

310C8 

38645  51 

39971 

68050 

55318 

21608958 

37357 

60372 

59061 

50 

53322 

82334 

70646 

254CO 

55184 

69703 

80707 

49 

66688 

96635 

85994 

41983 

73035 

89060 

•5001784 

48 

80068 

98  10952 

•07  01359 

58527 

90909 

•38  08444 

.22891 

47 

93464 

25286 

16743 

75091 

•27  08807 

27855 

44029 

46 

1-90  OC874 

39636 

32146 

91677 

26729 

47293 

65198 

45 

20299 

54003 

47567 

2-17  08283 

44674 

66758 

86398  44 

83738 

68387 

63007 

24911 

62643 

86250 

•51  07629  1  43 

47193 

82787 

78465 

41559 

80C36 

•39  057<:9 

28890  1  42 

60663 

97204 

93942 

58229 

98653 

25316 

50183 

41 

74147 

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•0809438 

74920 

•28  16093 

44889 

71507 

40 

87647 

26087 

24953 

91631 

34758 

64490 

92863  39 

1-91  01162 

40554 

40487 

2-18  08364 

52846 

84118 

•52  14249 

38 

14691 

55038 

56039 

25119 

70959 

•4003774 

35667  37 

28236 

69539 

71610 

41894 

89096 

23457 

57117 

36 

41795 

8405C 

87200 

58C91 

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43168 

78598  35 

55370 

98590 

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75510 

25442 

62906 

2-5300111  34 

68960 

•0013142 

18437 

92349 

43651 

82672 

21655  '  33 

82565 

27710 

34085 

2-19  09210 

61885 

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43231  i  32 

96186 

42295 

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26093 

80143 

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64839  |  31 

1-9209821 

56897 

65436 

42997 

98425 

42136 

86479!  30 

23472 

71516 

81140 

59923 

2-30  1(5732 

62013 

2-5408151 

29 

37138 

86153 

96864 

76871 

35064 

81918 

29855  28 

50819 

•01  00806 

•10  12607 

93840 

53420 

2-4201851 

51591  27 

64516 

15477 

28369 

2-20  10831 

71801 

21812 

73359  26 

78228 

30164 

44150 

27843 

90206 

41801 

95160|  25 

91956 

44869 

59951 

44878 

2-31  08637 

61819 

2-551C992I  24 

1-93  05699 

59592 

75771 

61934 

27092 

81864 

38S58  23 

19457 

74331 

91611 

79012 

45571 

2-4301938 

607561  22 

33231 

89088 

2-11  07470 

96112 

64076 

22041 

82686  21 

47020 

2-02  OS862 

23348  2-21  13234 

82606 

42172 

2-56  04649  20 

60825 

18654 

39246 

30379 

2-3201160 

62331 

26645  19 

74645 

33462 

55164 

47545 

19740 

82519 

48674  i  W 

88481 

48289 

71101 

64733 

38345 

2-44  02736 

7073&1  17 

1-9402333 

63133 

87057 

81944 

56975 

22982 

92830  i  16 

16200 

77994 

2-1203034    99177 

75630 

43256 

2-5714957;  15 

30083 

92873 

190302-2216432 

94311 

63559 

37118  14 

43981 

2-03  07769 

35046i    33709 

2-3313017 

83891 

59312  13 

57896 

22683 

51082i    51009 

31748 

2-45  04252 

81539  12 

71826 

37615 

67137 

68331 

50505 

24642 

2-58  03SOO  11 

85772 

52565 

83213    85676 

69287 

45061 

26094  10 

99733 

67532 

.  99308  2-23  03043 

88095 

65510 

48421   9 

1-9513711 

82517 

2-1315423    20433 

2-34  06928 

85987 

70782   8 

2770 

97519 

31559,    37845 

257  87 

2.46  06494 

9P177  i  7 

4171 

2-0412540 

477  14'    55280 

44672 

27030 

2-5915606   6 

5573 

2757 

63890    72738 

63582 

4759f 

38068   5 

6978 

42634 

80085    90218 

82519 

68191 

60564   4 

8383 

5770 

963012-2407721 

2-35  01481 

88816 

83095   3 

9791 

7280 

2-14125371    25247 

20469 

2-47  09470 

2-60  05659  i  2 

1-961200C 

8791 

28793    42796 

39483 

30155 

28258   1 

2610 

2-05  0303S 

45069    60368 

58524 

50869 

50891 

0 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

' 

NAT.  COTAN. 


NATURAL   TANGENTS. 


265 


69° 

70° 

71° 

72° 

73°    74° 

75° 

2-60  50891 

2-74  74774 

2-90  42109 

3-07  76835 

3-27  08526  3-48  74144 

3-73  20508 

73558 

99661 

6957C 

3-08  07325 

42588 

3-49  12470 

63980 

96259 

2-75  24588 

97089 

37869 

76715 

50874 

3-74  07546 

2-61  18995 

49554 

2-91  24649 

68468 

3-28  10907 

89356 

51207 

41766 

74561 

5225G 

92122 

451G4 

3-5027916 

949C3 

64571 

99608 

79909 

3-09  29831 

79487 

66555 

3-7538815 

87411 

2-76  24695 

2-9207610 

60596 

3-29  1J876 

3-51  05273 

82763 

2-62  10286 

49822 

35358 

91416 

48330 

44070 

3-7626807 

33196 

74990 

63152 

3-10  22291 

82851 

82946 

70947 

56141 

2-77  00199 

90995 

53223 

3-3017438 

3-5221902 

3-77  15185 

79121 

25448 

2-9318885 

84210 

52091 

60938 

59519 

2-6302136 

50738 

46822 

3-11  15254 

86811 

3-5300054 

3-78  03951 

25186 

76069 

74S07 

46353 

3-31  21598 

39251 

48481 

48271 

2-78  01440 

2-94  02840 

77509 

56452 

78528 

93109 

71392 

26853 

30921 

3-1208722 

91373 

3-54  17886 

3-7937835 

94549 

52307 

59050 

3U991 

3-32  26362 

57325 

82661 

2-64  17741 

77802 

87227 

71.317 

61419 

96846 

3-80  27585 

40969 

2-7903339 

2-95  15453 

3-13  02701 

96543 

3-55  36449 

72609 

64232 

28917 

43727 

34141 

3-3331736 

7G133 

3-81  17733 

87531 

54537 

72050 

65639 

66997 

3-5615900 

62957 

2-65  10867 

80198 

2-96  00422 

97194 

3-3402326 

55749 

3-8208281 

34238 

2-80  05901 

28842 

3-14  28807 

37724 

95C81 

53707 

57645 

31646 

57312 

60478 

73191 

3-57  35690 

99233 

81089 

57433 

85831 

92207 

3-3508728 

75794 

3-8344861 

2-6604569 

83263 

2-97  14399 

3-15  23994 

44333 

3-58  15975 

90591 

28085 

2-8109134 

43016 

55840 

80008 

56241 

3-843C424 

51638 

35048 

71683 

877M 

3-36  15753 

90590 

82358 

75227 

610(34 

2-98  00400 

316  19706 

51568 

3-5937024 

3-85  28396 

98853 

87003 

29167 

51728 

87453 

77543 

74537 

2-67  22516 

2-8213045 

57983 

83808 

3-37  23408 

3-6018146 

3-8620782 

46215 

39129 

86850 

3-17  15948 

59434 

58835 

67131 

69951 

65256 

2-9915766 

48147 

95531 

99609 

3-87  13584 

93725 

91426 

44734 

80406 

3-38  31G99 

3-61  404C9 

•   60142 

2-68  17535 

2-83  17639 

73751 

3-18  12724    67938 

81415 

3-8806S05 

413S3 

43896 

3-0002820 

451023-3904249 

3-6222447 

53574 

65267 

70196 

SI  939 

77540)   40631 

63560 

3-8900448 

89190 
2-6913149 

96539 
2-8422926 

61109 
90330 

3-19  10039    77085 
42598  3-40  13612 

3-6304771 
46064 

47429 
94516 

37147 

49356 

3-61  19603 

75217 

50210 

8744  1 

3-9041710 

61181 

75831 

48926 

3-2007897 

86882 

3-64  28911 

89011 

85254 

2-85  02349 

78301 

40638  3-41  23626 

704C7 

3-91  36420 

2-70  09364 

28911 

3-0207728 

73440    60443 

3-65  12111 

•  83937 

33513 

55517 

37207 

3-21  06304    97333 

53844 

3-9231563 

57699 

82168 

66737 

392283-4234297 

95665 

79297 

81923 

2-8608SG3 

96320 

72215|    71334 

3-6637575 

3-9327141 

2-71  06186 

35602 

3-03  25954 

3-22052633-4308446 

79575 

75094 

30487 

62386 

55641 

38373 

45631 

3-67  21665 

3-9423157 

54826 

89215 

85381 

71546 

82891 

63845 

71331 

79204 

2-87  160S8 

3-0415173 

3-23  04780  3-44  20226 

3-6806115 

3-95  19615 

2-72  03620 

43007 

45018 

38078    57635 

48475 

68011 

28076 

69970 

74915 

71438    95120 

90927 

3-96  16518 

52569 

96979 

3-05  04S66 

3-24  01860  3-45  32679 

0-6933469 

65137 

77102 

2-8824033 

34870 

38346)    70315 

76104 

3-97  13868 

2-73  01674 

51132 

6492S 

7  1  895  3-46  08026  3-70  1  8830 

62712 

262S4 

78277 

95038 

3-25  05508 

45813  1    61648 

3-9811669 

50934 

2-8905467 

3-06  25203 

39184 

83676  3-7104558 

60739 

75023 

32704 

55421 

729243-4721616 

47561 

3-9909924 

2-74  00352 

599SO 

85694 

3-26  06728  j    59632 

90658 

59223 

25120 

87314 

3-07  If  020 

405961    977263-7233847 

4-0008636 

49927 

2-90  14688 

4(3400 

745293-4835896!    77131 

58165 

74774 

42109 

76835 

3-27  08526 

74144  3-7320.508 

4-01  07809 

20° 

19° 

18° 

17° 

16° 

15° 

14° 

NAT.  COTAN. 


266 


NA TURAL    TANGENTS. 


/ 

76° 

77° 

78° 

79° 

80° 

81°  i  82°   ' 

0 

4-01  07809 

4-33  14759 

4-70  46301 

5-1  445540 

5'6  712818  63  137515'7'1  153697  60 

1 

57570 

72316 

4-71  136S6 

525557 

809446 

256601 

304190  59 

i 

4-020744U 

i-34  30018 

81256 

605813 

906394 

376126 

455308  58 

8 

57440 

878C6 

4-72  49012 

686311 

5-7  003663 

496092 

607056  57 

4 

4-03  07550 

4-35  45S61 

4-73  16J54 

167051 

101256 

616502 

759437  56 

6 

57779 

4-36  04003 

850f,3 

848035 

199173 

737369 

912456  55 

6 

4-0408125 

62293 

4-74  53401 

9292G4 

297416 

858665 

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7 

585,0 

4-37  20731 

4-75  21907 

5-2  010738 

395988 

980422 

.  220422  53 

8 

4-05  09174 

79317 

90603 

092459 

494889  6-4  102633 

375378  52 

9 

59877 

4-38  38054 

4-76  53490 

174428 

594122 

225301 

530987;  51 

10 

4-06  10700 

93940 

4-77  28568 

256647 

883688 

348428 

687255  50 

11 

61  043 

4-39  559^7 

97837 

339116 

793588 

472017 

844184  49 

VI 

4-07  12707 

4'40  15164 

4-78  C7300 

421836 

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596070 

7-3-001780  48 

13 

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74504 

4-79  30957 

504809 

994400 

720591 

160047  47 

14 

4-08  15199 

4-41  33996 

4-80  06808 

588035 

5-8095315 

845581 

318989  46 

15 

66627 

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70854 

671517 

196572 

971043 

478010  45 

ie 

4-09  18178 

4-42  53439 

4-81  47096 

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6-5  096981 

638916|  44 

17 

69852 

4-43  13392 

4-82  17536 

839251 

400117 

223396 

799909,  43 

18 

4-10  21649 

73500 

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923505 

502410 

350293 

961595!  42 

19 

73569 

4-4433762 

4-83  59010 

5-3  008018 

605051 

4776727-4123978!  41 

20 

4-1125614 

94181 

4-8430045 

092793 

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605538 

287064  40 

21 

77784 

4-45  54V50 

4-85  01282 

177830 

811386 

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4-12  30079 

4-4615489 

72719 

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23 

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4-86  44359 

348696 

5-9019138 

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24 

4-13  35046 

4:47  37428 

4-87  16201 

434527 

123550 

6-6  121919 

946514  36 

25 

87719 

98636 

88248 

520626 

228322 

252258 

7-5  11  3178  35 

2(1 

4-1440519 

4-48  60004 

4-88  60499 

»  606993 

333455 

383100 

280571 

34 

27 

93446 

4-49  21532 

4-89  3295*6 

693030 

438952 

514449 

448699 

33 

2* 

4-15  46501 

83221 

4-90  05620 

780538 

544815 

646307 

617567 

32 

29 

99885 

4-50  45072 

78491 

867718 

651045 

778677 

787179 

31 

30 

4-16  52998 

4-51  07085 

4-91  51570 

955172 

757644 

911562 

957541 

30 

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4-17  06440 

69261 

4-92  24859 

5-4  042901 

864614l6-7  044966 

7-6128657 

29 

32 

60011* 

4-5231601 

9S358 

130300 

971957 

178891 

300533 

28 

3:; 

4-18  13713 

94105 

4-93  72068 

219188 

6-0079076 

313341 

473174 

27 

34 

67546 

4-53  56773 

4-9445990 

307750 

187772 

448318 

646584 

26 

SO 

4-19  21510 

4-54  19608 

4-95  20125 

396592 

296247 

583826 

820769 

25 

86 

75600 

82608 

94474 

485715 

405103 

'  719867 

995735 

24 

37 

4-20  29835 

4-5545776 

4-90  69037 

575121 

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856446 

7-7  171486 

23 

3S 

84193 

4-5609111 

4-97  43817 

664812 

623967 

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348028 

22 

39 

4-21  38690 

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4-9818813 

754788 

733979 

6-8  131227 

525366 

21 

40 

93318 

4-57  36287 

94027 

845052 

844381 

269437 

703506 

20 

41 

4-22  48080 

4-58  00129 

4-90  69459 

935604 

955174 

408196 

882453 

19 

42 

4-23  02977 

64141 

5-0045111 

5-5  026446 

6-1  0663HO 

547508 

7-8062212 

18 

4:) 

58009 

4-59  28325 

5-01  20984 

117579 

177943 

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242790 

17 

44 

4-24  13177 

92680 

97078 

209005 

289923 

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16 

45 

68482 

4-60  57207 

5-02  73395 

300724 

402303 

968799 

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15 

46 

4-25  23923 

4-61  21908 

50349935 

392740 

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6-9  110359 

789489 

14 

47 

79501 

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5-04  26700 

485052 

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13 

48 

4-26  35218 

4-62  51832 

5-05  03090 

577663 

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7-9  158151 

12 

49 

91072 

4-63  17056 

80907 

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50 

4-27  47066 

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5-06  58352 

763786 

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51 

4-28  03199 

4-64  48034 

5-07  36025 

857302 

6-2085106 

826781 

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9 

52 

59472 

4-6513788 

5-08  13928 

951121 

200347 

971806 

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8 

53 

4-29  15885 

79721 

92061 

5-6045247 

316007  7-0  117441 

8-0  094835 

7 

51 

72440 

4-66  45832 

5-09  70426 

139680 

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284796 

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4-30  29136 

4-67  12124 

5-1049024 

23-4421 

548588   410482 

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5 

66 

85974 

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5-11  27855 

329474 

665515   557905 

667394 

4 

57 

4-31  42955 

4-68  45248 

5-12  OG921 

424838 

782868   705934 

860042 

3 

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4-32  00079 

4-69  12083 

86224 

520516 

9006511   854573 

8-1  053599 

2 

59 

57347 

79100 

5-13  65763 

616509 

6-30188687*1003826 

248071 

1 

60 

4-3314759 

4-70  46301 

5-14  45540 

712818 

137515 

153697 

443464 

0 

/ 

13° 

12° 

11° 

10° 

9° 

8° 

7° 

/ 

NAT.  COTAN. 


NATURAL    TANGENTS. 


267 


83° 

84°    85° 

86° 

87° 

88° 

89° 

/ 

8-1443464 

9-5143645  11.430052 

14-300066 

19-081137 

28-636253 

57-289962 

60 

639786 

410613   468474 

360696 

187930 

877089 

58-261174 

59 

837041 

679068   507154 

421230 

295922 

29-122006 

59-265872 

58 

8-2  03523U 

949022 

546093 

482273 

405133 

371100 

60-305820 

57 

234384 

9-6  220486 

685294 

543833 

515584 

124499 

61-382905 

56 

434485 

493475 

624761 

6U5916 

627296 

*•  8225  19 

02-499154  55 

635547 

708000 

664496 

668529 

740291 

30-144119 

63-656741 

54 

837579 

9-7  044075 

704500 

731679 

854591 

4115SO 

64-858008 

53 

8-3  040586 

321713 

7-14779 

795372 

970219 

6&3307 

66-105473 

52 

244577 

600927 

78533Q 

859616 

20-OS719U 

959928 

67-401854 

51 

449558 

881732 

8261C7 

924417 

205553 

31-241577 

68-750087 

50 

655536 

9-8  164140 

867282 

989784 

325308 

52S392 

70-153346 

49 

862519 

448166 

908682 

15-055723 

440486 

820516 

71-615070 

48 

8-4  070515 

733823 

950370 

122242 

509115 

32-118099 

73-138991 

47 

279531 

9-9  021125 

992349 

1S9349 

693220 

421295 

74-729165 

46 

4S9573 

310088 

12.034022 

257052 

818828 

730265 

76?390009 

45 

700651 

600724 

077192 

325358 

945966 

33-045173 

78-126342 

44 

912772 

893050 

120002 

394276 

21-074064 

360194 

79-943430 

43 

8-5  125943 

10-018708 

163236 

463814 

204949 

693509 

81-847041 

42 

340172 

048283 

206716 

533981 

336851 

34-027303 

83-843507 

41 

555468 

078031 

250505 

604784 

470401 

367771 

85-939791 

40 

771838 

107954 

294609 

076233 

605630 

715115 

88-143572 

39 

989290 

138054 

339028 

748337 

742569 

35-069546 

90-463336 

38 

8-6207833 

108332 

383768 

821105 

881251 

431282 

92-908487 

37 

427475 

198789 

428831 

804545 

22-021710 

800553 

95-489475 

36 

648223 

229428 

474221 

968667 

163980 

36-177596 

98-217943 

35 

870088 

260249 

519942  16-043482 

308097 

562659 

101-10690 

34 

8-7  093077 

291255 

565997 

118998 

454096 

950001 

104-17094 

33 

317198 

322447 

612390 

195225 

602015 

37-357892 

107-42648 

32 

542461 

353827 

659125 

272174 

751892 

76S613 

110-89205 

31 

768874 

385397 

706205 

349855 

903766 

38-188459 

114-58865 

30 

996440 

417158 

753634 

428279 

23-057077 

617738 

118-54018 

29 

8-8  2251S6 

449112 

801417 

507456 

213066 

39-056771 

122-77396 

28 

455103 

481261 

849557 

587396 

371777 

505895 

127-32134 

27 

686206 

513607 

898058 

668112 

532052 

905400 

132-21851 

26 

918505 

546151 

946924 

749614 

694537 

40-435837 

137-50745 

25 

8-9  152009 

578895 

.9961  CO 

831915 

859277 

917412 

143-23712 

24 

386726 

611841 

13-045769 

915025 

24-026320 

41-410588 

149-46502 

23 

622608 

644992 

095757 

998957 

195714 

915790 

156-25908 

22 

859843 

678348 

146127 

17-083724 

367509 

42-433464 

163-70019 

21 

9-0  098261 

711913 

196883 

109337 

541758 

964077 

171-88540 

20 

337933 

745/587 

248031 

255809 

718512 

43-508122 

180-93220 

19 

578867 

779673 

299574 

343155 

897826 

44-066113 

190-98419 

18 

821074 

813872 

351518 

431385 

25-079757 

638596 

202-21875 

17 

9-1  064564 

848288 

403867 

520516 

264361 

45-226141 

214-85762 

16 

309318 

882921 

456625 

610559 

451700 

8293.J1 

229-18166 

15 

555436 

917775 

509799 

701529 

641832 

40-448Sr>2 

245-55198 

14 

802838 

952850 

563391 

793442 

834823 

47-085343 

264-44080  - 

13 

9-2051564 

988150 

617409 

886310 

26-0307^6 

739501 

286-47773 

12 

301627 

11-023676 

671856 

980150 

229638 

48-412084 

312-52137 

11 

553035 

059431 

726738 

18-074977 

431000 

49-10P881 

343-77371 

10 

805802 

095416 

783060 

170807 

636090 

315726 

381-97099  I  9 

9-3  059936 

131635 

837827 

267054 

844984 

50-548500 

429-71757   8 

315450 

168089 

894045 

305537 

27-056557 

51-303157 

491-10600 

7 

572355 

204780 

950719 

464471 

271486 

52-080673 

572-95721 

6 

830663 

241712 

14-007856 

564473 

489853 

882109 

68^54887 

5 

9-4  090384 

278885 

065459 

605562 

711740 

53-708587 

859-43630 

4 

351531 

316304 

123686 

767754 

937233 

54-561300 

1145-9153 

3 

614116 

353970 

182092 

871068 

28-166422 

•>5-441517 

1718-8732 

2 

878149 

391885 

241134 

975523 

399397 

56-350590 

3437-7467 

1 

9.5  143645 

430052 

300666 

19-081137 

636253 

57-289962 

Infinite. 

0 

6° 

5° 

4° 

3° 

2° 

1° 

0° 

/ 

NAT.  COTAN. 


TABLE  XVI. 


CHORDS,  VERSED  SINES,  EXTERNAL  SECANTS,  AND 
TANGENTS  OF  A  ONE-DEGREE  CURVE. 

The  angles  of  the  table  are  the 
intersection  angles,  I,  equal  to 
the  total  central  angle  included 
between  the  tangent  points. 

To  find  the  corresponding  func- 
tion for  any  other  curve,  divide 
the  tabular  number  by  the  de- 
gree of  curvature. 

The  unit  chord  is  assumed  to 
be  one  hundred  feet  long. 

By  using  radius  of  5,730  feet, 
the  chord  column  of  the  table  can  be  made  serviceable  for 
plotting. 


270  CHORDS,   VERSED   SIXES,   EX  TERN  A  7.   SECANTS. 


55 

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^ 

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6   M   ro  invo  co   6   M   co  IOMD  06   6   *   ro  invo  co   6    M   ro  io\o  oo   6   M   ro  m^o  CO   Q     i 

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CHORDS,   VERSED  SINES,  EXTERNAL  SECANTS.  271 


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0    CN|    •"J-xO  CO    O    CM    -*NO  OO    O    CM 


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274  CHORDS,  VERSED  SINES,  EXTERNAL   SECANTS. 


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SLOPES,  FOR   TOPOGRAPHY. 


315 


TABLE     XV. 

SLOPES,    FOR    TOPOGRAPHY. 


Degrees. 

Vertical  Rise 
in  100  Feet 
Horizontal. 

Horizontal 
Distance 
to  a  Rise  of 
10  Feet. 

Degrees. 

Vertical  Rise 
in  TOO  Feet 
Horizontal. 

Horizontal 
Distance 
to  a  Rise  of 
10  Feet. 

i 

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572-9 

19 

34-43 

29.0 

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286.4 

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56-7 

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60 

173-20 

5-7 

TABLE   XVII. 


RISE  PER  MILE  OF  VARIOUS  GRADES. 


318 


RISE  PER  MILE  OF  VARIOUS   GRADES. 


Grade 
per 
Station. 

Rise  per 
Mile. 

Grade 

c  Per 
Station. 

Rise  per 
Mile. 

Grade 

c  Per 
Station. 

Rise  per 
Mile. 

Grade 
per 
Station. 

Rise  per 
Mile. 

.01 

.528 

.61 

32.208 

I.  21 

63.888 

1.81 

95-568 

.02 

i  .056 

.62 

32-736 

I  .22 

64.416 

1.82 

f       /• 
96  .  096 

•03 

1.584 

.63 

33-264 

1.23 

64.944 

1.83 

96.624 

.04 

2.  112 

.64 

33-792 

1.24 

65-472 

1.84 

97.152 

•05 

2.640 

•65 

34-320 

I-25 

66.000 

1.85 

97.680 

.06 

3.168 

.66 

34-848 

1.26 

66.528 

1.86 

98.208 

.07 

3.696 

.67 

35.376 

1.27 

67.056 

1.87 

98.736 

,08 

4.224 

.68 

35'9°4 

1.28 

67.584 

1.88 

99.264 

.09 

4-752 

.69 

36.432 

1.29 

68.112 

1.89 

99.792 

.10 

5.280 

.70 

36.960 

1.30 

68.640 

i.  go 

100.320 

.11 

5.808 

•7i 

37.488 

*•$» 

69.168 

1.91 

100.848 

.12 

6.336 

•72 

38  -Ol6 

1.32 

69.696 

1.92 

101.376 

•13 

6.864 

•73 

38.544 

!-33 

70.224 

*-93 

101.904 

.14 

7.392 

•74 

39.072 

1-34 

70.752 

1.94 

102.432 

•!5 

7.920 

•75 

39.600 

i«35 

.71.280 

i-95 

102.960 

.16 

8.448 

.76 

40.  128 

1.36 

71.808 

1.96 

103.488 

•I? 

8.976 

•77 

40.656 

J-37 

72-336 

1.97 

104.016 

.18 

9-504 

.78 

41.184 

1.38 

72.864 

1.98 

104.544 

.19 

IO.O32 

•79 

41.712 

i>39 

73.392 

1.99 

105.072 

.20 

10.560 

.80 

42.240 

i.4o 

73.920 

2.OO 

105.600 

.21 

I  I.  088 

.81 

42.768 

1.41 

74.448 

2.10 

110.880 

.22 

II  .6l6 

.82 

43-296 

1.42 

74.976 

2.20 

116.160 

•23 

12.144 

.83 

43.824 

1.43 

75-504 

2.30 

121.440 

.24 

12.672 

.84 

44-352 

1.44 

76.032 

2.40 

126.720 

.25 

13.200 

•85 

44.880 

1.45 

76.560 

2.50 

132.000 

.26 

13.728 

.86 

45.408 

1.46 

77.088 

2.6o 

137.280 

•A 

I4.256 
14.784 

.87 
.88 

45.936 
46.464 

1.47 
1.48 

77.616 
78.144 

2.70 
2.8o 

142.560 
147.840 

.29 

15.31-2 

.89 

46.992 

1.49 

78.672 

2.90 

153.120 

•30 

15.840 

.90 

47.52o 

1.50 

79.200 

3-00 

158.400 

•31 

16.368 

.91 

48.048 

1-51 

79.728 

3-10 

163.680 

•32 

16.896 

.92 

48.576 

1.52 

80.256 

3.20 

168.960 

•33 

17.424 

•93 

49.104 

«»53 

80.784 

3-30 

174.240 

•34 

17.952 

•94 

49.632 

J-54 

81.312 

3-40 

179.520 

•35 

18.480 

•95 

50  .  i  60 

i>55 

81.840 

3-50 

184.800 

•36 

19.008 

.96 

50.688 

1.56 

82.368 

3-60 

190.080 

•37 

I9.536 

•97 

51.216 

»-57 

82.896 

3-7° 

195.360 

.38 

20.064 

.98 

5!-744 

1.58 

83.424 

3-80 

200.640 

•39 

20.592 

•99 

52.272 

1-59 

83.952 

3-9° 

205  .  920 

.40 

21.120 

1.  00 

52.800 

i.  60 

84.480 

4.00 

211.200 

.41 

21.648 

.01 

53-328 

1.61 

85.008 

4.10 

216.480 

.42 

22.  176 

.02 

53-856 

1.62 

85-536 

4.20 

221  .760 

•43 

22.704 

•°3 

54.384 

1.63 

86.064 

4-30 

227.040 

•44 

23.232 

.04 

54-912 

1.64 

86.592 

4.40 

232.320 

•45 

23.760 

•05 

55.440 

1.65 

87.120 

4-5° 

237.600 

.46 

24.288 

.06 

55.968 

1.66 

87.648 

4.60 

242.880 

•47 

24.816 

.07 

56.496 

1.67 

88.176 

4.70 

248.160 

.48 

25.344 

.08 

57-024 

1.68 

88.704 

4.80 

253.440 

•49 

25.872 

.09 

57.552 

1.69 

89.232 

4.90 

258.720 

•5° 

26.400 

.10 

58.080 

1.70 

89.760 

5.00 

264.000 

•51 

26.928 

.11 

58.608 

1.71 

90.288 

5.10 

269.280 

•52 

27.456 

.  12 

59-l36 

1.72 

90.816 

5.20 

274.560 

•53 

27.984 

•13 

59-664 

i-73 

9J-344 

5-30 

279.840 

•54 

28.512 

.14 

60.  192 

1.74 

91.872 

5-40 

285.120 

•55 

29.040 

•!5 

60.720 

1  .  j  r 

92.400 

5-5° 

290.400 

•56 

29.568 

.16 

61.248 

1.76 

92.928 

5-6o 

295.680 

•57 

30.096 

•!? 

61.776 

1.77 

93-456 

5-7° 

3OO  .  960 

•58 

30.624 

.18 

62.304 

1.78 

93-984 

5-80 

306.240 

•59 

S1'^2 

I.I9 

62.832 

1.79 

94-512 

5-9° 

311.520 

.60 

31.680 

I  .20 

63.360 

i.  80 

95.040 

6.00 

316.800 

INDEX. 


PAGE 

Abbreviations  explained ix 

Acres,  roods,  and  perches  in  square  feet,  Table  VI 152 

Adjustment  and  use  of  instruments 23 

Angles  of  frogs,  to  find 129 

index,  to  find 69 

intersection,  to  find 55 

plane 12 

to  read  on  verniers 43 

tangential  and  deflection .  50 

of  switch-rails 130 

Apex  distance  of  curves,  to  find 52 

Arc,  functions  of,  to  find 13 

Arithmetical  complement 6 

Axemen,  duties  of 84 

Azimuths  of  North  Star,  Table  III 150 

Barometer,  levelling  by 29 

Bench-marks,  proper  intervals  for 83 

Bubble,  to  adjust  on  level 25 

to  adjust  on  transit 40 

Chain,  to  lay  out  curves  with 63 

Chainman,  duties  of 82 

Chief  engineer,  duties  of 79 

Chords,  to  calculate 54,58 

Table  XVI 269 

Circle,  propositions  concerning 49 

Circular  arcs  to  radius  of  1,  Table  VII 152 

Complement  of  an  angle 12 

arithmetical 6 

Compound  curves.     See  Curves. 

Contour  maps,  utility  of 85 

Correction  for  curvature  and  refraction  in  levelling 28 

Cosines  defined 12 

Crossings,  plain  rules  for  laying  off 139 

Cross-hairs,  to  adjust 24,  26,  40 

eccentricity  of " • 24 

to  put  in  new 44 

319 


320  INDEX. 

PAGE 

Cross-sectioning.    See  Slope  stakes. 

Cubes  and  cube  roots  of  numbers,  Table  XI 161 

Curves,  circular,  on  railroad  defined 51 

to  find  radius,  length,  degree,  apex  distance,  chord,  versed  sine, 

and  external  secant 52,  56 

form  for  field  notes 70 

Curves,  how  to  lay  out  on  the  ground,  — 

with  the  chain  only 63 

with  transit  and  chain 66 

hints  as  to  field-work 82 

protractor  for 84 

slackening  grade  on 87 

terminal 88 

Curves,  simple,  location  of,  — 

how  to  proceed  when  the  P.  C.  is  inaccessible 93 

to  shift  the  P.  C.  in  order  to  strike  a  fixed  tangent 96 

to  change  radius  from  same  P.  C.  in  order  to  strike  a  fixed  tan- 
gent    97 

to  triangulate  on 94 

to  pass  through  a  fixed  point 127, 128 

Curves,  compound,  — 

how  to  proceed  when  the  P.  C.  C.  is  inaccessible 95 

to  compound  a  curve  in  order  to  strike  a  fixed  tangent    ....  98 

to  shift  a  P.  C.  C.  in  order  to  strike  a  fixed  tangent 99 

summary  of  rules  for 101 

to  compound  into  a  tangent  intersecting  main  curve  on  concave 

side 102 

to  compound  into  a  tangent  intersecting  main  curve  on  convex 

side   .     .     . ' 103 

Curves,  reversed,  — 

parallel  tangents,  radii  equal ,  115 

parallel  tangents,  radii  unequal      .     .     .     .    ! 117_ 

angles  unequal,  radii  equal 119 

angles  unequal,  tangent  points  fixed,  radii  equal 120 

divergent  tangents,  radii  equal,  advancing  towards  intersection     .  123 

receding  from  intersection 124 

to  shift  a  P.  R.  C.  in  order  to  strike  a  fixed  tangent 125 

Curves,  miscellaneous,  — 

elevation  of  outer  rail    . 141,142 

degree  of,  to  find  by  calculation 52,  55 

to  find  on  ground 145,  146 

to  connect  curves  of  contrary  flexure  by  short  tangents      ...  89 

to  locate  a  Y  from  a  tangent 103 

from  a  convex  curve 104 

from  a  concave  curve 106 

to  locate  a  tangent  to  a  curve  from  a  fixed  point 108 

to  two  curves  already  located 109 

to  substitute  a  curve  for  a  tangent  connecting  two  curves    .    .    .  109 

terminal  curves 88 


INDEX.  321 

PAGE 

Curves,  miscellaneous  —  continued. 

trackmen's  table  of  curves  and  spring  of  rails    . 143 

vertical  curves,  to  calculate 36 

to  project 39 

Datura  in  levelling 27 

Decimals  of  an  acre  per  100  feet  for  various  widths,  Table  V 151 

Deflection  angles  and  distances  explained 50 

to  find 57,  64,  68 

short  rule  for  sub-deflections 68 

limit  in  field-practice 82 

Degree  of  curve,  to  calculate 52,  55 

to  find  on  ground 145,  146 

Deviations  from  project  admissible  on  location 81 

Distances,  tangential  and  deflection,  defined 50 

table  of 155 

of  frogs  from  toe  of  switch 130,  132 

tables  of 135,  136 

Elevation  of  outer  rail  on  curves 141 

table  of 142 

Excavation  and  embankment,  to  stake  out 30 

External  secants,  to  find 54 

of  a  1°  curve,  Table  XVI 269 

Extreme  elongations  of  North  Star,  Table  II 150 

Feet  in  decimals  of  a  mile,  Table  VIII 153 

Field-work,  suggestions  concerning 79,  85 

Field-book,  form  of,  for  level 27 

for  transit 70 

for  slope  stakes 33,  34,  35 

Frogs  and  switches    .    . 129 

rules  for  angles  and  distances    .    .    .    ., 130 

table  of,  switch-rails  straight 135 

switch-rails  curved 136 

plain  rules  for  locating,  switch-rails  straight 132 

switch-rails  curved 133 

on  narrow  gauges 134 

patterns  for 134 

Functions,  trigonometrical,  defined 12 

logarithmic,  of  arcs,  to  find 14 

General  propositions  in  trigonometry 15 

as  to  circles 49 

Grade,  to  slacken  on  curves 87 

rise  per  mile,  Table  XVII. 317 

Grade  lines,  how  to  project  on  map 86 

how  to  trace  in  field  .....  81 


Heights,  to  find  by  barometer  and  thermometer 


322  INDEX. 


PAGE 

Inches  in  decimals  of  a  foot,  Table  IX 153 

Index  angles,  to  determine 69 

Instruments,  adjustment  and  use  of 23 

Intersection  angles  of  tangents,  to  find 55 

desirable  to  fix  on  ground 66 

Level,  to  adjust 24 

Leveller,  duties  of ~    .  83 

Levelling,  art  of 26 

by  barometer  and  thermometer 29 

correction  for  curvature  and  refraction 28 

form  for  field-book 27 

rules  for  exact  work 27 

rules  for  survey  and  location 28 

suggestions  concerning 83 

Location,  problems  in  field 94 

admissible  errors  on  ground 81 

form  of  record  for „ 81 

projects,  hints  concerning 84 

of  terminal  curves 88 

of  a  Y • 103,  104, 106 

Logarithms  explained 3 

multiplication  by 5 

division  by 6 

of  numbers,  to  find 4 

Table  XII 179 

roots  and  powers  by 7 

Logarithmic  sines,  tangents,  &c.,  to  find 13 

table  of,  XIII 197 

Maps,  contour,  utility  of 85 

notes  for  .....  ^ 82,  83 

not  sufficient  for  intelligent  projects 79 

Meridian,  to  establish 44 

by  equal  shadows 45 

by  North  Star 45 

times  of  passage  of  North  Star,  Table  I. 149 

Multiplication  by  logarithms 5 

Natural  sines,  tangents,  &c.,  defined 12 

Table  XIV 243,256 

Needle,  magnetic,  to  adjust 41 

to  re-magnetize 44 

hints  as  to  management 44 

bearings  should  always  be  noted 82 

North  Star,  to  establish  meridian  by 45 

times  of  meridian  passage,  Table  1 149 

extreme  elongations  of,  Table  II 150 

azimuths  and  natural  tangents,  Table  III 150 


INDEX.  323 

PAGE 

Obstacles  in  the  field  to  vision 71 

to  measurement 73 

Ordinates  of  circular  curves,  to  find 58,  59 

of  parabola,  to  4ind 36 

of  a  r  curve,  Table  X 155 

Parabola,  ordinates  of 36,  59 

Plane  trigonometry 12 

Powers  and  roots  of  numbers  by  logarithms 7 

Propositions,  general,  in  trigonometry 15 

Protractor  for  curves  described 84 

how  to  make 85 

Rails,  table  of  spring  for  trackmen 143 

Radius  of  a  curve,  how  to  find 52,  54,  56 

of  a  turnout  curve 129 

plain  rule  for,  on  curves 133 

for  narrow  gauges 134 

Radii  and  their  logarithms,  Table  X 155 

Records,  forms  for 81 

Refraction  and  curvature,  correction  for 28 

Reversed  curves.     See  Curves. 

Rise  per  mile  of  various  grades,  Table  XVII 317 

Rod,  levelling 28 

how  to  read 42 

Rodman,  duties  of 83 

Roods  and  perches  in  decimals  of  an  acre,  Table  iV 151 

Roots  and  powers  of  numbers  by  logarithms 7 

Senior  assistant,  duties  of 80 

equipment  for 81 

Sines  defined 12 

Shadows,  to  fix  true  north  by 44 

Slopes  for  topography,  Table  XV 315 

Slopeman,  duties  of 84 

Slope  stakes,  to  set £0 

for  earth  excavation 31 

for  embankment 33 

for  hillsides  and  rock 35 

field  record  of  work 34 

Spring  of  rails,  table  for  trackmen 143 

Squares,  cubes,  and  roots  of  numbers,  Table  XI.    . 161 

Supplement  of  an  angle .' 12 

Survey,  form  for  record 81 

to  facilitate 82 

Switch-rails,  angles  of _    .    .    .  130 

tables  of 135,136 

Tangent,  or  apex  distance  of  curve,  to  find 52,  54 


324  INDEX. 

PAGE 

Tangent  of  a  1°  curve,  Table  XVI 269 

to  curve  from  a  fixed  point,  how  to  locate 108 

to  two  curves  on  the  ground,  how  to  locate 109 

Tangential  angles  and  distances  explained 50 

how  to  find 57,58,64 

Thermometer,  levelling  by 29 

Track  problems 115 

Trackmen's  plain  rules  for  finding  frog  distances 132,  133 

tables  of  turnouts      . 135,  136 

plain  rules  for  laying  off  turnouts  with  tape-measure  and  pins    .      137 

crossings  on  straight  lines  and  on  curves 139 

elevation  of  outer  rail 142 

instructions  how  to  put  in  missing  stakes  on  curves  with  tape- 
measure     144 

table  of  curves  and  spring  of  rails 143 

explanation  of  the  trackmen's  tables 144 

how  to  find  the  degree  of  a  curve 145, 146 

Transit,  adjustment  of 40 

cross-hairs 24 

Transitman,  duties  of 82 

Triangles,  solution  of,  — 

two  angles  and  a  side  given 16 

two  sides  and  an  angle  given 17 

three  sides  given 18 

Triangles,  right-angled,  solution  of 19 

Trigonometry,  plane 12 

general  propositions 15 

Turnouts.    See  Trackmen. 

Vernier  explained 42 

on  transit 43 

Versed  sines  defined 12 

to  calculate 54,  58 

of  a  1' curve,  Table  XVI  . 269 

Vertical  curves,  to  calculate 36 

to  project 39 


TABLES  : 

Ordinates  of  a  1°  curve 60 

For  locating  terminal  curves 88 

Tangents  between  curves  of  contrary  flexure 89 

Turnouts,  switch-rails  straight 135 

switch-rails  curved 136 

Elevation  of  outer  rail  on  curves 142 

Curves  and  spring  of  rails 143 

I.    Time  of  meridian  passage  of  North  Star 149 

II.    Time  of  extreme  elongations  of  North  Star     .......  150 


INDEX.  325 

PAGE 

HE.    Azimuths  of  North  Star,  and  their  natural  tangents    ....  150 

IV.    Roods  and  perches  in  decimal  parts  of  an  acre 151 

V.    Decimals  of  an  acre  in  one  chain  length  of  100  feet,  and  of 

various  widths 151 

VI.    Acres,  roods,  and  perches  in  square  feet 152 

VII.    Circular  arcs  to  radius  of  1 152 

VIII.    Feet  in  decimals  of  a  mile 153 

IX.    Inches  reduced  to  decimal  parts  of  a  foot 153 

X.    Radii  and  their  logarithms,  middle  ordinates,  and  deflection 

distances 155 

XI.    Squares,  cubes,  roots,  and  reciprocals  of  numbers,  from  1  to 

1,042 161 

XII.    Logarithms  of  numbers  from  1  to  10,000 179 

XIII.  Logarithmic  sines,  cosines,  tangents,  and  cotangents  ....  197 

XIV.  Natural  sines  and  cosines 243 

Natural  tangents  and  cotangents 256 

XVI.    Chords,  versed  sines,  external  secants,  and  tangents  of  a  1° 

curve 269 

XV.    Slopes  for  topography 315 

XVII.    Rise  per  mile  of  various  grades 817 


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A  Treatise  on  the  Strength  of  Bridges  and  Roofs.  Comprising 
the  determination  of  algebraic  formulas  for  strains  in  horizontal, 
inclined  or  rafter,  triangular,  bowstring,  lenticular,  and  other 
trusses,  from  fixed  and  moving  loads.  With  practical  applications 
and  examples,  for  the  use  of  students  and  engineers.  Eighty- 
seven  woodcut  illustrations.  Second  edition.  8vo,  cloth  .  .  5  00 

MERRILL  (W.  E.). 

Iron  Truss  Bridges  for  Railroads.  The  method  of  calculating 
strains  in  trusses,  with  a  careful  comparison  of  the  most  promi- 
nent trusses,  in  reference  to  economy  in  combination,  &c.  Illus- 
trations. 4to,  cloth 5  00 

8 


D.  VAN  NO  STRAND'S  SCIENTIFIC  BOOKS. 

STUART  (C.  B.). 

The  Civil  and  Military  Engineers  of  America.  With  nine  finely 
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gravings  of  some  of  the  most  important  works  constructed  in 
America.  8vo,  cloth  .  .  . $5  00 

CLARK  (D.  K.). 

Fuel ;  its  Combustion  and  Economy.  Embracing  portions  of  the 
well-known  works  of  C.  WYE  WILLIAMS,  "Combustion  of  Coal 
and  the  Prevention  of  Smoke,"  and  of  T.  T.  PRIDEAUX'S  work  on 
"  The  Economy  of  Fuel."  With  extensive  additions  on  Recent 
Practice  in  the  Combustion  and  Economy  of  Fuel,  Coal,  Coke, 
Wood,  Peat,  Petroleum,  &c.  12mo,  cloth 1  50 


A  Manual  of  Rules,  Tables,  and  Data  for  Mechanical  Engineers. 

Large  8vo.    1,012  pages.    Illustrated.    Cloth 7  50 

Half  morocco 10  00 

WILLIAMSON  (R.  S.). 

On  the  Use  of  the  Barometer  on  Surveys  and  Reconnoissances. 
Part  I.  Meteorology  in  its  Connection  with  Hypsometry.  Part  II. 
Barometric  Hypsometry.  With  illustrative  tables  and  engravings. 

4to,  cloth 15  00 

Practical  Tables  in  Meteorology  and  Hypsometry,  in  connection 
with  the  use  of  the  Barometer.  4to,  cloth 2  50 

CLEVENGER. 

A  Treatise  on  the  Method  of  Government  Surveying,  as  prescribed 
by  the  U.  S.  Congress  and  Commissioner  of  the  General  Land 
Office;  with  complete  Mathematical,  Astronomical,  and  Practical 
Instructions  for  the  use  of  the  United  States  Surveyors  in  the 
Field.  By  S.  V.  CLEVENGEE.  Morocco 2  50 

RICHARDS'S  INDICATOR. 

A  Treatise  on  the  Richards  Steam-Engine  Indicator,  with  an 
Appendix  by  F.  W.  BACON,  M.  E.  Third  edition,  revised  and  en- 
larged. 18mo,  flexible,  cloth 1  00 

BOW  (R.  H.). 

A  Treatise  on  Bracing,  with  its  application  to  Bridges  and  other 
Structures  of  Wood  or  Iron.  One  hundred  and  fifty-six  illustra- 
tions. 8vo,  cloth  .  .'.".'.' 1  50 

HAMILTON  (W.  G.). 

Useful  Information  for  Railway  Men.  Sixth  edition,  revised  and 
enlarged.  562  pages.  Pocket  form,  morocco,  gilt  .  .  .  .  2  00 

STUART  (B.). 

How  to  become  a  Successful  Engineer.  Being  Hints  to  Youths 
intending  to  adopt  the  profession.  Sixth  edition.  12mo,  boards  .  50 


D.  VAN  NOSTRAN&S  SCIENTIFIC  BOOKS. 

SCRIBNER  (J.  M.). 

Engineers'  and  Mechanics'  Companion.  Comprising  United-States 
Weights  and  Measures,  Mensuration  of  Superficies  and  Solids, 
Tables  of  Squares  and  Cubes,  Square  and  Cube  Roots,  Circumfer- 
ence and  Areas  of  Circles,  The  Mechanical  Powers,  Centres  of 
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Bodies,  Strength,  Weight,  and  Crush  of  Materials,  Water  Wheels, 
Hydrostatics,  Hydraulics,  Statics,  Centres  of  Percussion  and  Gyra- 
tion, Friction  Heat,  Tables  of  the  Weight  of  Metals,  Scantling, 
&c.,  Steam  and  the  Steam-Eugine.  Eighteenth  edition,  revised. 
Morocco $1  50 


Engineers',  Contractors',  and  Surveyors'  Pocket  Table-Book. 
Comprising  Logarithms  of  Numbers,  Logarithmic  Signs  and  Tan- 
gents, Natural  Signs  and  Natural  Tangents,  the  Traverse  Table, 
and  a  full  and  complete  set  of  Excavation  and  Embankment  Ta- 
bles, together  with  numerous  other  valuable  tables  for  Engineers, 
&c.  Tenth  edition,  revised.  Morocco 1  50 

SCHUMANN  (F.). 

A  Manual  of  Heating  and  Ventilation  in  its  Practical  Application, 
for  the  use  of  Engineers  and  Architects.  Embracing  a  series  of 
tables  and  formulae  for  dimensions  of  heating,  flow  and  return 
pipes,  for  steam  and  hot-water  boilers,  flues,  &c.  Illustrated. 
Morocco 1  50 

SHIELDS  (J.  E.). 

Embracing  Discussions  of  the  Principles  involved  and  Descrip- 
tions of  the  Material  employed  in  Tunnelling,  Bridging,  Canal  and 
Road  Building,  &c 1  50 

MORRIS  (ELWOOD). 

Easy  Rules  for  the  Measurement  of  Earthworks,  by  means  of  the 
Prismoidal  Formula.  Illustrated.  8vo,  cloth 1  50 

GILLMORE  (Gen.  Q.  A.). 

Treatise  on  Limes,  Hydraulic  Cements,  and  Mortars.  With  numer- 
ous illustrations.  1  vol.  8vo,  cloth 4  00 

HARRISON   (W.  B.). 

The  Mechanic's  Tool  Book,  with  Practical  Rules  and  Suggestions 
for  use  of  Machinists,  Iron- Workers,  and  others.  Illustrated. 
12mo,  cloth 1  50 

HENRICI  (OLAUS). 

Skeleton  Structures,  especially  in  their  application  to  the  Building 
of  Steel  and  Iron  Bridges.  With  folding^plates  and  diagrams. 
8vo,  cloth 1  50 

HEWSON  (WILLIAM). 

Principles  and  Practice  of  Embanking  Lands  from  River  Floods, 

as  applied  to  the  Levees  of  the  Mississippi.    8vo,  cloth  .       .        .    2  00 

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HOLLEY  (A.  LJ. 

Railway  Practice.  American  and  European  Railway  Practice,  in 
the  Economical  Generation  of  Steam,  including  the  Materials  and 
Construction  of  Coal-burning  Boilers,  Combustion,  the  Variable 
Blast,  Vaporization,  Circulation,  Superheating,  Supplying  and 
Heating  Feed-water,  &c.,  and  the  Adaptation  of  Wood  and  Coke 
burning  Engines  to  Coal-burning;  and  in  Permanent  Ways  includ- 
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&c.  With  seventy-seven  lithographed  plates.  Folio,  cloth  »  $12  00 

MINIFIE  (WILLIAM). 

Mechanical  Drawing.  A  Text-Book  of  Geometrical  Drawing  for 
the  use  of  Mechanics.  With  illustrations  for  Drawing  Plans,  Sec- 
tions, and  Elevations  of  Railways  and  Machinery;  an  Introduction 
to  Isometrical  Drawing,  and  an  Essay  on  Linear  Perspective  and 
Shadows.  Illustrated.  Ninth  edition.  With  an  Appendix  on  the 
Theory  and  Application  of  Colors.  8vo,  cloth 4  00 

STILLMAN  (PAUL). 

Steam-Engine  Indicator,  and  the  improved  Manometer  Steam  and 
Vacuum  Gauges,  —  their  Utility  and  Application.  New  edition. 
12mo,  flexible  cloth 1  00 

WALKER  (W.  H.). 

Screw  Propulsion.  Notes  on  Screw  Propulsion  :  its  Rise  and  His- 
tory.  8vo,  cloth ...75 

MAC  CORD  (C.  W.). 

A  Practical  Treatise  on  the  Slide-Valve  by  Eccentrics,  examining 
by  methods  the  action  of  the  Eccentric  upon  the  Slide- Valve,  and 
explaining  the  practical  processes  of  laying  out  the  movements 
adapting  the  valve  for  its  various  duties  in  the  steam-engine.  For 
the  use  of  engineers,  draughtsmen,  machinists,  and  students  of 
valve-motions  in  general.  Illustrated.  4to,  cloth  .  .  .  .  3  00 

KIRKWOOD  (JAMES  P.). 

Report  on  the  Filtration  of  River  Waters  for  the  Supply  of  Cities, 
as  practised  in  Europe.  Made  to  the  Board  of  Water  Co«&mi8- 
sioners  of  the  city  of  St.  Louis.  Illustrated  by  thirty  double-plate 
engravings.  4to,  cloth 15  00 

BURGH  (N.  P.). 

Modern  Marine  Engineering,  applied  to  Paddle  and  Screw  Propul- 
sion. Consisting  of  36  colored  plates,  259  practical  woodcut  illus- 
trations, and  403  pages  of  descriptive  matter;  the  whole  being  an 
exposition  of  the  present  practice  of  James  Watt  &  Co.,  J.  &  G. 
Rennie,  R.  Napier  &  Sons,  and  other  celebrated  firms.  Thick  4to 

vol.    Cloth 25  00 

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SIMMS  (F.  WJ. 

A  Treatise  on  the  Principles  and  Practice  of  Levelling,  showing 
its  application  to  purposes  of  Railway -Engineering  and  the  Con- 
struction of  Roads,  &c.  Illustrated.  8vo,  cloth  .  .  .  .  $2  50 

BURT  (W.  A.). 

Key  to  the  Solar  Compass,  and  Surveyor's  Companion.  Compris- 
ing all  the  rules  necessary  for  use  in  the  field.  Second  edition, 
pocket-book  form,  tuck  . 2  50 

THE  PLANE-TABLE. 

Its  Uses  in  Topographical  Surveying.  From  the  papers  of  the 
United-States  Coast  Survey.  Illustrated.  8vo,  cloth  .  .  .  2  00 

HOWARD  (C.  R.). 

Earthwork  Mensuration  on  the  Basis  of  the  Prismoidal  Formula. 
Containing  simple  and  labor-saving  method  of  obtaining  pris- 
moidal  contents  directly  from  end  areas.  Illustrated  by  examples, 
and  accompanied  by  plain  rules  for  practical  uses.  Illustrated. 
8vo,  cloth 1  50 

GRUNER  (M.  L.). 

The  Manufacture  of  Steel.  Translated  from  the  French  by  LENOX 
SMITH.  With  an  Appendix  on  the  Bessemer  process  in  the  United 
States,  by  the  translator.  Illustrated  by  lithographed  drawings 
and  woodcuts.  8vo,  cloth 3  50 

BARBA  (J.). 

The  Use  of  Steel  for  Constructive  Purposes ;  Method  of  Working, 
Applying,  and  Testing  Plates  and  Brass.  With  a  Preface  by  A.  L. 
HOLLEY,  C.E.  12mo,  cloth 1  50 

DUBOIS  (A.  J.). 

The  New  Method  of  Graphical  Statics.    With  sixty  illustrations. 

8vo,  cloth         . 1  50 

EDDY  (H.  T.). 

Researches  in  Graphical  Statics,  embracing  New  Constructions  in 
Graphical  Statics,  a  new  General  Method  in  Graphical  Statics, 
and  the  Theory  of  Internal  Stress  in  Graphical  Statics.  8vo,  cloth,  1  50 

WEALE'S   RUDIMENTARY  SCIENTIFIC  SERIES. 

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over  two  hundred  distinct  works  in  almost  every  department  of 
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4®~  General  Catalogue  of  Scientific  Books,  with  Index  to  Authors,  76 
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